From the Classical Beta Distribution to Generalized Beta Distributions

Home , Arcsine distribution, Beta prime distribution, Burr distribution, Hyperbolic distribution, Rayleigh distribution, Triangular distribution

FROM THE CLASSICAL BETA DISTRIBUTION

TO GENERALIZED BETA DISTRIBUTIONS

Title

A project submitted to the School of Mathematics, University of Nairobi in partial fulfillment of the requirements for the degree of Master of Science in Statistics. By Otieno Jacob I56/72137/2008

Supervisor: Prof. J.A.M Ottieno

School of Mathematics University of Nairobi July, 2013

FROM THE CLASSICAL BETA DISTRIBUTION TO GENERALIZED BETA DISTRIBUTIONS

Half Title

Declaration

This project is my original work and has not been presented for a degree in any other University

Signature ______

Otieno Jacob

This project has been submitted for examination with my approval as the University Supervisor

Signature ______

Prof. J.A.M Ottieno

iii

Dedication

I dedicate this project to my wife Lilian; sons Benjamin, Vincent, John; daughter Joy and friends. A special feeling of gratitude to my loving Parents, Justus and Margaret, my aunt Wilkista, my Grandmother Nereah, and my late Uncle Edwin for their support and encouragement.

Acknowledgement

I would like to thank my family members, colleagues, friends and all those who helped and supported me in writing this project. My sincere appreciation goes to my supervisor, Prof. J.A.M Ottieno. I would not have made it this far without his tremendous guidance and support. I would like to thank him for his valuable advice concerning the presentation of this work before the Master of Science Project Committee of the School of Mathematics at the University of Nairobi and for his encouragement and guidance throughout. I would also like to thank Dr. Kipchirchir for his questions and comments which were beneficial for finalizing the project.

I would like to recognize the financial support from the Higher Education Loans Board (HELB) without which this study might not have been accomplished. Most of all, I would like to thank the Almighty God for always being there for me.

Table of Contents Title...... i

Half Title ...... ii

Declaration ...... iii

Dedication ...... iv

Acknowledgement ...... v

Table of Contents ...... 1

Executive Summary...... 16

1 Chapter I: General Introduction ...... 18

1.1 Introduction ...... 18

1.2 Problem Statement ...... 19

1.3 Objective...... 19

1.4 Literature Review ...... 20

2 Chapter II: The Classical Beta Distribution and its Special Cases ...... 27

2.1 Beta and Gamma Functions...... 27

2.2 Construction of the Classical Beta Distribution ...... 31

2.2.1 From the beta function ...... 31

2.2.2 From stochastic processes ...... 32

2.2.3 From ratio transformation of two independent gamma variables ...... 32

2.2.4 From the rth order statistic of the Uniform distribution ...... 33

2.3 Properties of the Classical Beta Distribution ...... 34

2.3.1 rth-order moments ...... 34

2.3.2 The first four moments ...... 34

2.3.3 Mode ...... 36

2.3.4 Variance ...... 37

2.3.5 Skewness ...... 37

2.3.6 Kurtosis ...... 39

2.4 Shapes of Classical Beta Distribution ...... 41

2.5 Applications of the Classical Beta Distribution ...... 45

2.6 Special Cases of the Classical Beta Distribution ...... 45

2.6.1 Power Distribution Function ...... 45

2.6.2 Properties of power distribution ...... 46

2.6.3 Shape of Power distribution ...... 47

2.6.4 Uniform Distribution Function ...... 48

2.6.5 Properties of the Uniform distribution ...... 48

2.6.6 Shape of Uniform distribution ...... 50

2.6.7 Arcsine Distribution Function...... 51

2.6.8 Properties of the arcsine distribution ...... 51

2.6.9 Shape of Arcsine distribution ...... 54

2.6.10 Triangular shaped distributions (a=1, b=2) ...... 54

2.6.11 Properties of the Triangular shaped distribution (a=1, b=2) ...... 54

2.6.12 Shape of triangular distribution (a=1, b=2) ...... 57

2.6.13 Triangular shaped distributions (a=2, b=1) ...... 57

2.6.14 Properties of the Triangular shaped distribution (a=2, b=1) ...... 57

2.6.15 Shape of triangular distribution (a=2, b=1) ...... 59

2.6.16 Parabolic shaped distribution ...... 60

2.6.17 Properties of parabolic shaped distribution ...... 60

2.6.18 Shape of Parabolic distribution ...... 62

2.7 Transformation of Special Case of the Classical Beta Distribution ...... 62

2.7.1 Wigner Semicircle Distribution Function ...... 62

2.7.2 Properties of Wigner Semicircle ...... 63

2.7.3 Shape of Wigner semicircle distribution ...... 65

3 Chapter III: Type II Two Parameter Inverted Beta Distribution ...... 66

3.1 Construction of Beta Distribution of the Second Kind ...... 66

3.2 Properties of Beta Distribution of the Second Kind ...... 67

3.2.1 rth-order moments ...... 67

3.2.2 Mean ...... 69

3.2.3 Mode ...... 70

3.2.4 Variance ...... 71

3.2.5 Skewness ...... 72

3.2.6 Kurtosis ...... 73

3.2.7 Shapes of Beta distribution of the second kind ...... 75

3.3 Application of Beta Distribution of the Second Kind ...... 75

3.4 Special Cases of Beta Distribution of the Second Kind...... 75

3.4.1 Lomax (Pareto II) distribution function ...... 75

3.4.2 Properties of Lomax distribution...... 76

3.4.3 Shape of Lomax distribution ...... 77

3.4.4 Log-Logistic (Fisk) distribution function ...... 77

3.4.5 Properties of Log-logistic distribution ...... 78

3.4.6 Shape of Log-logistic distribution ...... 79

3.5 Limiting Distribution ...... 79

4 Chapter IV: The Three Parameter Beta Distributions ...... 80

4.1 Introduction ...... 80

4.2 Case I: Libby and Novick Three Parameter Beta Distribution ...... 80

4.3 Case II: McDonald’s Three Parameter Beta I Distribution ...... 83

4.4 Gamma Distribution as a Limiting Distribution ...... 84

4.5 Case III ...... 85

4.5.1 3 Parameter Beta Distribution Derived from the Beta Distribution of the Second Kind .... 85

4.5.2 Case IV ...... 86

4.5.3 Case V ...... 86

4.5.4 Case VI ...... 87

5 Chapter V: The Four Parameter Beta Distributions ...... 88

5.1 Introduction ...... 88

5.2 Generalized Four Parameter Beta Distribution of the First Kind ...... 88

5.2.1 Properties of generalized beta distribution of the first kind ...... 90

5.3 Applications of Generalized Beta Distribution of the First Kind ...... 93

5.4 Special Cases of Generalized Beta Distribution of the First Kind ...... 93

5.4.1 Pareto type 1 distribution function ...... 94

5.4.2 Properties of Pareto type 1 distribution ...... 94

5.4.3 Stoppa (generalized Pareto type 1) distribution function ...... 97

5.4.4 Properties of Stoppa distribution ...... 98

5.4.5 Kumaraswamy distributions ...... 100

5.4.6 Properties of Kumaraswamy distribution ...... 101

5.4.7 Generalized Power distribution Function ...... 103

5.4.8 Properties of power distribution ...... 103

5.4.9 Generalized Gamma Distribution Function ...... 105

5.4.10 Unit Gamma Distribution Function ...... 106

5.5 Generalized Four Parameter Beta Distribution of the Second Kind ...... 107

5.5.1 Properties of generalized beta distribution of the second kind ...... 108

5.6 Special Case of the Generalized Beta Distribution of the Second Kind...... 113

5.6.1 Singh-Maddala (Burr 12) Distribution ...... 113

5.6.2 Properties of Singh-Maddala distribution ...... 114

5.6.3 Shapes of Singh-Maddala distribution...... 117

5.6.4 Generalized Lomax (Pareto type II) Distribution ...... 118

5.6.5 Properties of Generalized Lomax distribution ...... 118

5.6.6 Inverse Lomax Distribution ...... 121

5.6.7 Properties of Inverse Lomax distribution ...... 122

5.6.8 Dagum (Burr type 3) Distribution ...... 123

5.6.9 Properties of Dagum distribution ...... 123

5.6.10 Para-logistic distribution ...... 125

5.6.11 General Fisk (Log-Logistic) Distribution Function ...... 125

5.6.12 Properties of Fisk (log-logistic) distribution ...... 126

5.6.13 Fisher (F) Distribution Function ...... 127

5.6.14 Properties of F distribution ...... 127

5.6.15 Half Student’s t distribution ...... 129

5.6.16 Logistic Distribution Function ...... 130

5.6.17 Generalized Gamma Distribution Function ...... 130

5.6.18 Properties of Generalized Gamma distribution ...... 131

5.6.19 Gamma Distribution Function ...... 134

5.6.20 Properties of Gamma distribution...... 134

5.6.21 Inverse Gamma distribution ...... 136

5.6.22 Properties of Inverse Gamma distribution ...... 137

5.6.23 Weibull Distribution Function ...... 140

5.6.24 Properties of Weibull distribution ...... 140

5.6.25 Exponential distribution Function ...... 141

5.6.26 Properties of exponential distribution ...... 142

5.6.27 Chi-Squared distribution function ...... 143

5.6.28 Properties of Chi-squared distribution ...... 143

5.6.29 Rayleigh distribution function ...... 145

5.6.30 Properties of Rayleigh distribution ...... 145

5.6.31 Maxwell distribution function ...... 146

5.6.32 Properties of Maxwell distribution...... 147

5.6.33 Half Standard normal function ...... 149

5.6.34 Properties of Half Standard normal distribution...... 149

5.6.35 Half-normal distribution function ...... 150

5.6.36 Properties of Half-normal distribution ...... 151

5.6.37 Half Log-normal distribution function ...... 152

5.6.38 Properties of Half Log-normal distribution ...... 152

5.7 Four Parameter Generalized Beta Distribution Function ...... 153

5.7.1 Properties of the four parameter generalized beta distribution ...... 154

6 Chapter VI: The Generalized Beta Distribution Based on Transformation Technique ...... 155

6.1 Introduction ...... 155

6.2 Five Parameter Generalized Beta Distribution Function Due to McDonald ...... 155

6.3 Properties of the Five Parameter Generalized Beta Distribution ...... 157

6.3.1 rth–order moments ...... 157

6.3.2 Mean ...... 158

6.3.3 Variance ...... 158

6.4 Special Cases of the Five Parameter Generalized Beta Distribution Function ...... 158

6.4.1 The four parameter generalized beta distribution of the first kind ...... 158

6.4.2 The four parameter generalized beta distribution of the second kind ...... 159

6.4.3 The four parameter generalized beta distribution...... 159

6.4.4 The four parameter generalized Logistic distribution function ...... 159

6.4.5 The three parameter Inverse Beta distribution of the first kind ...... 160

6.4.6 The three parameter Stoppa (generalized pareto type I) distribution function ...... 160

6.4.7 The three parameter Singh-Maddala (Burr12) distribution function ...... 160

6.4.8 The three parameter Dagum (Burr3) distribution function ...... 161

6.4.9 The two parameter Classical Beta distribution ...... 161

6.4.10 The two parameter Pareto type I distribution function ...... 161

6.4.11 The two parameter Kumaraswamy distribution function ...... 162

6.4.12 The two parameter generalized Power distribution function ...... 162

6.4.13 The two parameter Beta distribution of the second kind ...... 162

6.4.14 The two parameter Inverse Lomax distribution function ...... 163

6.4.15 The two parameter General Fisk (Log-Logistic) distribution function ...... 163

6.4.16 The two parameter Fisher (F) distribution function ...... 163

6.4.17 The one parameter Power distribution function ...... 164

6.4.18 The one parameter Wigner semicircle distribution function ...... 164

6.4.19 The one parameter Lomax (pareto type II) distribution function ...... 165

6.4.20 The one parameter half student’s (t) distribution function...... 165

6.4.21 Uniform distribution function ...... 165

6.4.22 Arcsine distribution function ...... 166

6.4.23 Triangular shaped distribution functions...... 166

6.4.24 Parabolic shaped distribution functions ...... 166

6.4.25 Log-Logistic (Fisk) distribution function ...... 167

6.4.26 Logistic distribution function ...... 167

6.5 Construction of the Five Parameter Weighted Generalized Beta Distribution Function...... 169

6.6 Properties of Five Parameter Weighted Generalized Beta Distribution ...... 170

6.6.1 rth-order moments ...... 170

6.6.2 Mean ...... 170

6.6.3 Variance ...... 170

6.6.4 Skewness ...... 171

6.6.5 Kurtosis ...... 171

6.7 Special Cases of the Five Parameter Weighted Generalized Beta Distribution Function ...... 172

6.7.1 Weighted beta of the second distribution function ...... 172

6.7.2 Weighted Singh-Maddala (Burr12) distribution function ...... 172

6.7.3 Weighted Dagum (Burr3) distribution function ...... 172

6.7.4 Weighted Generalized Fisk (Log-Logistic) distribution function ...... 173

6.8 Construction of the Five Parameter Exponential Generalized Beta Distribution Function ...... 175

6.9 Properties of Five Parameter Exponential Generalized Beta Distribution ...... 176

6.9.1 Moment-generating function ...... 176

6.9.2 Mean of the exponential generalized beta distribution ...... 177

6.10 Special Cases of Exponential Generalized Beta Distribution ...... 177

6.10.1 Exponential generalized beta of the first kind ...... 178

6.10.2 Exponential generalized beta of the second kind ...... 178

6.10.3 Exponential generalized gamma ...... 179

6.10.4 Generalized Gumbell distribution ...... 179

6.10.5 Exponential Burr 12 ...... 179

6.10.6 Exponential power distribution function ...... 180

6.10.7 Two parameter exponential distribution function ...... 180

6.10.8 One parameter exponential distribution function ...... 180

6.10.9 Exponential Fisk (logistic) distribution ...... 181

6.10.10 Standard logistic distribution ...... 181

6.10.11 Exponential weibull distribution ...... 181

7 Chapter VII: Generalized Beta Distributions Based on Generated Distributions ...... 184

7.1 Introduction ...... 184

7.2 Beta Generator Approach ...... 184

7.3 Various Beta Generated Distributions ...... 185

7.3.1 Beta-Normal Distribution - (Eugene et. al (2002)) ...... 185

7.3.2 Beta-Exponential distribution – (Nadarajah and Kotz (2006)) ...... 186

7.3.3 Beta-Weibull Distribution – (Famoye et al. (2005)) ...... 188

7.3.4 Beta-Hyperbolic Secant Distribution – (Fischer and Vaughan. (2007)) ...... 189

7.3.5 Beta-Gamma – (Kong et al. (2007)) ...... 189

7.3.6 Beta-Gumbel Distribution – (Nadarajah and Kotz (2004)) ...... 190

7.3.7 Beta-Frèchet Distribution (Nadarajah and Gupta (2004)) ...... 191

7.3.8 Beta-Maxwell Distribution (Amusan (2010)) ...... 192

7.3.9 Beta-Pareto Distribution (Akinsete et.al (2008))...... 193

7.3.10 Beta-Rayleigh Distribution (Akinsete and Lowe (2009)) ...... 194

7.3.11 Beta-generalized-logistic of type IV distribution (Morais (2009)) ...... 195

7.3.12 Beta-generalized-logistic of type I distribution (Morais (2009)) ...... 196

7.3.13 Beta-generalized-logistic of type II distribution (Morais (2009)) ...... 196

7.3.14 Beta-generalized-logistic of type III distribution (Morais (2009)) ...... 197

7.3.15 Beta-beta prime distribution (Morais (2009))...... 197

7.3.16 Beta-F distribution (Morais (2009)) ...... 198

7.3.17 Beta-Burr XII distribution (Paranaíba et.al (2010)) ...... 198

7.3.18 Beta-Dagum distribution (Condino and Domma (2010)) ...... 200

7.3.19 Beta-(fisk) log logistic distribution ...... 201

7.3.20 Beta-generalized half normal Distribution (Pescim, R.R, et.al (2009)) ...... 201

7.4 Beta Exponentiated Generated Distributions...... 203

7.4.1 Exponentiated generated distribution ...... 203

7.4.2 Exponentiated exponential distribution (Gupta and Kundu (1999)) ...... 204

7.4.3 Exponentiated Weibull distribution (Mudholkar et.al. (1995)) ...... 204

7.4.4 Exponentiated Pareto distribution (Gupta et.al (1998))...... 205

7.4.5 Beta exponentiated generated distribution ...... 205

7.4.6 Generalized beta generated distributions ...... 205

7.4.7 Generalized beta exponential distribution (Barreto-Souza et al. (2010)) ...... 206

7.4.8 Generalized beta-Weibull distribution ...... 207

7.4.9 Generalized beta-hyperbolic secant distribution ...... 208

7.4.10 Generalized beta-normal distribution ...... 208

7.4.11 Generalized beta-log normal distribution ...... 209

7.4.12 Generalized beta-Gamma distribution ...... 210

7.4.13 Generalized beta-Frèchet distribution ...... 210

8 Chapter VIII: Generalized Beta Distributions Based on Special Functions ...... 212

8.1 Introduction ...... 212

8.2 Confluent Hypergeometric Function ...... 212

8.3 The Differential Equations ...... 213

8.4 Gauss Hypergeometric Function...... 217

8.5 The Differential Equations ...... 217

8.6 Appell Function ...... 220

8.7 Nadarajah and Kotz’s Power Beta Distribution ...... 221

8.8 Compound Confluent Hypergeometric Function ...... 221

8.9 Beta-Bessel Distribution (A.K Gupta and S.Nadarajah (2006)) ...... 222

9 Chapter IX: Summary and Conclusion ...... 224

9.1 Summary...... 224

9.2 Conclusion ...... 225

10 Appendix ...... 226

A.1 Classical Beta Distribution ...... 226

A.2 Power Distribution (b=1 in table A.1) ...... 226

A.3 Uniform Distribution (a=b=1 in table A.1) ...... 227

A.4 Arcsine Distribution ...... 227

A.5 Triangular Shaped (a=1, b=2 in table A.1) ...... 228

A.6 Triangular Shaped (a=2, b=1 in table A.1) ...... 228

A.7 Parabolic Shaped...... 228

A.8 Wigner Semicircle ...... 229

A.9 Beta Distribution of the Second Kind ...... 229

A.10 Lomax (Pareto II) (a=1 in table A.9)...... 230

A.11 Log Logistic (Fisk) (a=b=1 in table A.9) ...... 230

A.12 Libby and Novick (1982) Beta Distribution ...... 231

A.13 McDonald’s (1995) beta distribution of the first kind...... 231

A.14 Three Parameter Beta Distribution ...... 231

A.15 Chotikapanich et.al (2010) Beta of the Second Kind Distribution ...... 231

A.16 General Three Parameter Beta Distribution ...... 231

A.17 Three Parameter Beta Distribution ...... 232

A.18 Generalized Four Parameter Beta Distribution of the First Kind ...... 232

A.19 Pareto Type I (b=1, p=-1 in table A.1) ...... 233

A.20 Stoppa (Generalized Pareto Type I) (a=1, p=-p in table A.1) ...... 234

A.21 Kumaraswamy (a=1, q=1 in table A.1) ...... 235

A.22 Generalized Power (b=1, p=1 in table A.1) ...... 236

A.23 Generalized Four Parameter Beta Distribution of the Second Kind ...... 237

A.24 Singh Maddala (Burr 12) ...... 238

A.25 Generalized Lomax (Pareto II) ...... 239

A.26 Inverse Lomax (b=1, p=1 in table A.23) ...... 240

A.27 Dagum (Burr 3) (b=1 in table A.23) ...... 240

A.28 Para-Logistic (a=1, b = p in table A.23) ...... 241

A.29 General Fisk (Log –Logistic) ...... 241

A.30 Fisher (F) ...... 241

A.31 Half Student’s t ...... 242

A.32 Logistic ...... 242

A.33 Generalized Gamma ...... 242

A.34 Gamma ...... 243

A.35 Inverse Gamma ...... 243

A.36 Weibull ...... 244

A.37 Exponential ...... 244

A.38 Chi-Squared ...... 244

A.39 Rayleigh ...... 245

A.40 Maxwell ...... 245

A.41 Half Standard Normal ...... 245

A.42 Half Normal ...... 246

A.43 Half Log Normal ...... 246

A.44 Four Parameter Generalized Beta Distribution ...... 246

A.45 Five Parameter Generalized Beta Distribution ...... 247

A.46 Five Parameter Weighted Generalized Beta Distribution ...... 248

A.47 Weighted Beta of the Second Kind ...... 248

A.48 Weighted Singh-Maddala (Burr12) ...... 248

A.49 Weighted Dagum (Burr3) ...... 248

A.50 Weighted Generalized Fisk (Log-Logistic) ...... 249

A.51 Five Parameter Exponential Generalized Beta Distribution ...... 249

A.52 Exponential Generalized Beta Distribution of the First Kind ...... 249

A.53 Exponential Generalized Beta Distribution of the Second Kind ...... 249

A.54 Exponential Generalized Gamma (Generalized Gompertz) ...... 250

A.55 Generalized Gumbell ...... 250

A.56 Exponential Power ...... 250

A.57 Exponential (Two Parameter) ...... 250

A.58 Exponential (One Parameter) ...... 251

A.59 Exponential Fisk (Logistic)...... 251

A.60 Standard Logistic ...... 251

A.61 Exponential Weibull ...... 251

A.62 Beta Generator ...... 251

A.63 ith Order Statistics (a=i, b=n-i+1 in table A.62) ...... 252

A.64 Beta-Normal (Substituting G(X) and g(X) in table A.62) ...... 252

A.65 Beta-Exponential (Substituting G(X) and g(X) in table A.62) ...... 252

A.66 Beta-Weibull (Substituting G(X) and g(X) in table A.62) ...... 252

A.67 Beta-Hyperbolic Secant (Substituting G(X) and g(X) in table A.62) ...... 253

A.68 Beta-Gamma (Substituting G(X) and g(X) in table A.62) ...... 253

A.69 Beta-Gumbel (Substituting G(X) and g(X) in table A.62) ...... 253

A.70 Beta-Frèchet (Substituting G(X) and g(X) in table A.62) ...... 254

A.71 Beta-Maxwell (Substituting G(X) and g(X) in table A.62) ...... 254

A.72 Beta-Pareto (Substituting G(X) and g(X) in table A.62) ...... 254

A.73 Beta-Rayleigh (Substituting G(X) and g(X) in table A.62) ...... 255

A.74 Beta-Generalized-Logistic of Type IV (Substituting G(X) and g(X) in table A.62) ...... 255

A.75 Beta-Generalized-Logistic of Type I (Substituting q=1 in table A.74) ...... 255

A.76 Beta-Generalized-Logistic of Type II (Substituting q=1 in table A.74) ...... 255

A.77 Beta-Generalized-Logistic of Type III (Substituting q=1 in table A.74) ...... 255

A.78 Beta-Beta Prime ...... 256

A.79 Beta-F (Substituting G(X) and g(X) in table A.62) ...... 256

A.80 Beta-Burr XII (Substituting G(X) and g(X) in table A.62) ...... 256

A.81 Beta-Dagum (Substituting G(X) and g(X) in table A.62) ...... 256

A.82 Beta-Fisk (Log-Logistic) (Substituting G(X) and g(X) in table A.62) ...... 257

A.83 Beta-Generalized Half Normal (Substituting G(X) and g(X) in table A.62) ...... 257

A.84 Generalized Beta Generator ...... 257

A.85 Exponentiated Exponential (Substituting G(X) and g(X) in table A.84) ...... 257

A.86 Exponentiated Weibull (Substituting G(X) and g(X) in table A.84) ...... 258

A.87 Exponentiated Pareto (Substituting G(X) and g(X) in table A.84) ...... 258

A.88 Generalized Beta Exponential (Substituting G(X) and g(X) in table A.84) ...... 258

A.89 Generalized Beta Weibull (Substituting G(X) and g(X) in table A.84) ...... 258

A.90 Generalized Beta Hyperbolic Secant (Substituting G(X) and g(X) in table A.84) ...... 259

A.91 Generalized Beta Normal (Substituting G(X) and g(X) in table A.84) ...... 259

A.92 Generalized Beta Log Normal (Substituting G(X) and g(X) in table A.84) ...... 259

A.93 Generalized Beta Gamma (Substituting G(X) and g(X) in table A.84) ...... 259

A.94 Generalized Beta Frèchet (Substituting G(X) and g(X) in table A.84) ...... 260

A.95 Confluent Hypergeometric ...... 260

A.96 Gauss Hypergeometric ...... 260

A.97 Appell ...... 260

11 References ...... 262

List of Figures

Fig.1. 1: General Framework based on transformation ...... 21 Fig.1. 2: Beta Generated Distributions ...... 24 Fig.1. 3: Generalized Beta Distributions Based on Special Functions ...... 26 Fig.2. 1 : Left Skewedness ...... 42 Fig.2. 2: Case of beta unimodal distribution with 푎 > 0 ...... 43 Fig.2. 3: Case of beta unimodal distribution with 푏 > 0 ...... 43 Fig.2. 4: j-Shaped beta distribution ...... 44 Fig.2. 5: inverted j-Shaped beta distribution ...... 44

Fig.2. 6: Power distribution ...... 48 Fig.2. 7: Uniform distribution on the interval (0,1) ...... 50 Fig.2. 8: Arcsine distribution ...... 54 Fig.2. 9: Triangular shaped distribution...... 57 Fig.2. 10: Trangular shaped distribution ...... 59 Fig.2. 11: Parabolic distribution ...... 62 Fig.2. 12: Wigner semicircle distribution ...... 65 Fig.3. 1: Beta distribution of the second kind ...... 75 Fig.3. 2: Lomax distribution ...... 77 Fig.3. 3: Log-logistic distribution...... 79

Executive Summary

This Master‘s project considers on one hand, the construction of the two parameter classical beta distribution on the domain (0,1) and its generalization and on the other hand, the construction of the extended two parameter beta distribution on the domain (0,∞) and its generalization. It provides a comprehensive mathematical treatment of these two parameter beta distributions, their constructions, properties, shapes and special cases.

There are various ways of constructing distributions in general. Four ways of constructing the classical beta distribution considered in this work are constructions from,

i. The definition of the beta function; ii. A Poisson process; iii. The transformation of a ratio of two independent gamma variables; and iv. Order statistic.

Properties derived include the expressions of the rth moments, first four moments, mode, coefficient of skewness and coefficient of kurtosis. Special cases such as Power, Uniform, Arcsine, Triangular shaped, Parabolic shaped and Wigner Semicircle distributions are given. By the transformation technique a two parameter inverted beta distribution was obtained and its special cases such as Lomax (Pareto II) and Log-logistic (Fisk) distributions were constructed.

The work also looked at three methods of generalizing the classical beta distribution i.e. through:

1. Transformation technique 2. Generator Approach 3. Use of Special functions

The transformation technique resulted into the following:

i. Three parameter beta distributions of both first and second kinds. In particular, the Libby- Novick and McDonald‘s distributions. ii. Generalized four parameter beta distributions of the first and second kinds. Particular cases are the McDonald four parameter beta distribution of the first kind with its special

cases, the McDonald four parameter beta distribution of the second kind with its special cases and the four parameter generalized beta distribution. iii. The five parameter generalized beta distribution due to McDonald and Xu (1995).

Generalized distributions based on generator approach due to the work of Eugene et al. (2002) are classified into:

 Beta generated distributions  Exponentiated generated distributions  Generalized beta generated distributions

The generalized beta distributions based on the use of special functions considered includes the Confluent hypergeometric distribution and the Gauss Hypergeometric distribution from the Confluent hypergeometric function and the Gauss Hypergeometric function respectively. Other distributions are based on the Appell function and the Bessel function.

Tables showing special cases of beta distributions and their generalizations are given in the appendix.

1 Chapter I: General Introduction

1.1 Introduction

In statistical distributions, a vast activity has been observed in generalizing classical distributions by adding more parameters to make them more flexible in analyzing empirical data. Various methods have been developed to achieve this purpose. Among such methods are the transformation technique, generator approach, mixture (compounding) and use of special functions. In this work we examine the case of classical beta distribution. The term generalized used in this work has the notion of adding more parameters to the existing distribution in order to give a more universal distribution as opposed to the well known existing generalized distributions.

Historically, the work on beta distributions started as early as 1676 when Sir Isaac Newton wrote a letter to Henry Oldenberg as given by Dutka, J., (1981). In his letter

th 푥 dated 24 October, 1676, Newton evaluated 0 푦푑푥 as a series. The particular case of the result was applicable to the evaluation of the classical beta distribution. The two parameter classical beta distributions are among the celebrated Pearson system of statistical distributions derived by Karl Pearson in the 1890s in connection with his work on evolution.

A handbook of beta distribution and its applications was published in 2004 edited by Gupta and Nadarajah. The book enumerates the properties of beta distributions and related mathematical notions. It demonstrates the applications in the fields of economics, quality control, soil science and biomedicine. It further discusses the centrality of beta distributions in Bayesian inference, the beta-binomial model and applications of the beta- binomial distribution. However, immediately after that publication, further work has been done in that area. There is therefore a need to re-examine the works on beta distribution.

1.2 Problem Statement

Recent developments focus on new techniques for building meaningful distributions, a remarkable development has been observed in generalizing the classical beta distribution. There has been a growing interest in the applications of these distributions in the analysis of empirical data and for many statistical procedures. However, their applications are limited in important ways as they could not fit data very well. The main trend is therefore to add more parameters to make them more flexible in order to fit the existing empirical data.

1.3 Objective

The objective of this project is to review various methods that have been used to construct the beta distribution from the classical to generalized beta distributions. The specific objectives are:  to study the classical two parameter beta distribution within (0,1) domain and (0,∞) domain  to generalize the two parameter beta distributions using i. transformation technique ii. generator approach iii. some special functions

1.4 Literature Review

This section reviews various methods that have been used in constructing beta distributions. There are three classes of generalized beta distributions:

i. Those based on transformations

The generalized beta distributions based on transformation has largely been contributed by McDonald (1984). McDonald and his associates contributed in the development of the generalized beta distributions as an income distribution and in unifying the various research activities in closely related fields. McDonald and Xu (1995) introduced a five parameter beta distribution which nest the generalized beta and gamma distributions and include more than thirty distributions as limiting or special cases. McDonald and Richards (1987a, b) presented a four parameter generalized beta distribution which includes as special cases three and two parameter beta, generalized gamma, Weibull, power function, Pareto, lognormal, half-normal, uniform and others. Libby and Novick (1982) proposed a three parameter generalized beta distribution for utility fitting. The figure 1.1 below illustrates the transformation from the two parameter classical beta distribution to the five parameters generalized beta distribution where the transformations are given as follows:

T1: is the transformation changing a beta type I (classical) distribution to a two parameter beta type II (Inverted Beta) distribution

T2: transforms a 2 parameter to 3 parameter beta distributions

T3: transforms a 3 parameter to a 4 parameter beta distributions

T4: transforms a 4 parameter to 5 parameter beta distributions Ti‘s are defined in the coming chapters

T Beta Type I 1 Beta Type II

2 Parameters 2 Parameters

T2 T2

Generalized beta II Generalized beta I

3 Parameters 3 Parameters

T3 T3

Generalized beta I Generalized beta II

4 Parameters 4 Parameters

T4 T4

Generalized beta

5 Parameters

Fig.1. 1: General Framework based on transformation

ii. Generator approach based on CDF

The latest work involving the class of generalized beta distributions was introduced by Eugene et.al (2002) that defined the beta-normal distribution and studied some of its properties. Since then, several authors have been studying particular cases of this class of distributions. An advantage of the beta-normal distribution over the normal distribution is that the beta normal can be unimodal and bimodal. The bimodal region for the beta-normal distribution was studied by Famoye et. al (2003). Gupta and Nadarajah (2004) later derived a more general expression for the nth moment of the beta-normal distribution. Nadarajah and Kotz (2004) defined the beta-Gumbel distribution, which has greater tail flexibility than the Gumbel distribution. Nadarajah and Gupta (2004) defined the beta-Frèchet distribution and Barreto-Souza et.al. (2008) presented some additional mathematical properties. Nadarajah and Kotz (2006) defined the beta-exponential distribution and gave a detailed presentation. Famoye et.al (2005) defined the beta-Weibull and Lee et.al (2007) made some applications to censored data. Kong et.al (2007) proposed the beta-gamma distribution. The four parameter beta-pareto distribution was defined and studied by Akinsete et al (2008). Akinsete and Lowe (2009) proposed the beta-Rayleigh distribution. Amusan (2010) defined the beta-Maxwell distribution and studied some of its properties. The figure 1.2 below illustrates how the generator approach has been applied in the two parameters classical beta distribution to derive other generalized beta distributions.

iii. Generalized beta generated Barreto-Souza et al. (2010) proposed the generalized beta exponential distribution and discussed maximum likelihood estimation of its parameters.

iv. Beta-exponentiated distributions Since 1995, the exponentiated distributions have been widely studied in statistics with the development of various classes of these distributions. Mudholkar et.al. (1995) proposed the exponentiated Weibull distribution. Gupta et al. (1998) introduced the exponentiated Pareto distribution. Gupta and Kundu (1999) introduced the exponentiated exponential distribution as a generalization of the standard exponential distribution.

Original Original Classical beta distribution

Generalized Order Statistics Beta Normal Jones (2004) Eugene et al. (2002) Beta Gumbel Beta Frèchet Kotz and Nadarajah (2004) Nadarajah and Gupta (2004) Bimodal beta normal Beta Weibull Famoye et al (2003) Famoye & Olumolade (2005) Properties of Beta Frèchet Barreto-Souza et.al (2008)

Application on Beta Weibull nth moment of beta normal Lee et al. (2007 Gupta and Nadarajah (2004) Beta exponential Kotz and Nadarajah (2006)

Beta Rayleigh Beta Hyperbolic Secant Akinsete and Lowe (2009) Fischer and Vaughan (2007)

Beta Generalized Logistic of Type IV Beta Gamma Morais (2009) Kong et.al (2007) Beta Generalized Logistic of Type I

Beta Pareto Morais (2009) Akinsete et.al (2008) Beta F Beta Burr XII Morais (2009)

Beta Generalized Half Normal Paranaiba et.al (2010) Pescim R.R et.al (2009) Beta Beta Prime Beta (Fisk) Log Logistic Morais (2009) Paranaiba et.al (2010) Beta Dagum Beta Maxwell Condino and Domma (2010) Amusan (2010)

Fig.1. 2: Beta Generated Distributions

v. Based on special functions

Beta function is a special function. Other special functions have been used to generalize the beta distribution. Armero and Bayarri (1994) suggested Gauss hypergeometric function distribution in connection with marginal prior/posterior distribution. They obtained the generalized beta distribution by dividing the classical beta distribution by certain algebraic function. Nadarajah and Kotz (2004) also studied the beta distribution based on the Gauss hypergeometric function. Gordy (1988a) introduced confluent hypergeometric distribution and applied it to auction theory. He (Gordy (1988b)) also introduced the compound confluent hypergeometric distribution which contains McDonald and Xu‘s generalized beta, Gauss hypergeometric and confluent hypergeometric as special cases. Gordy generalized the classical beta distribution using the confluent hypergeometric function in the way Armero and Bayarri (1994) used Gauss hypergeometric function to generalize the beta to the Gauss hypergeometric distribution.

Nadarajah and Kotz (2006) introduced 퐹1 beta distribution based on Appell function of the first kind. The distribution is very flexible and contains several of the known generalization of the classical beta distribution as particular cases. The figure 1.3 below illustrates the relationship between the two parameters classical beta distribution and some of the special functions.

Fig.1. 3: Generalized Beta Distributions Based on Special Functions

Classical beta distribution

Gauss hypergeometric Confluent Armero/Bayarri (1994) hypergeometric function Gordy (1988a)

Compound confluent hypergeometric function Gordy (1988b)

Generalized beta Gauss Confluent McDonald/Xu (1995) hypergeometric hypergeometric

Appell Function, 퐹1 Beta Bessel Nadarajah/Kotz (2006) Gupta and Nadarajah (2006)

2 Chapter II: The Classical Beta Distribution and its Special Cases

2.1 Beta and Gamma Functions

Beta distributions are based on gamma functions. A brief description of these two functions is therefore in order.

i. Beta Function The beta function is a special function denoted by 퐵 푎, 푏 and defined as

ퟏ 퐵 푎, 푏 = 푥푎−1(1 − 푥)푏−1푑푥 , 푎 > 0, 푏 > 0 ퟎ

ii. Gamma Function The gamma function is a special function denoted by Γ and defined as

∞ 훤 푎 = 푡푎−1푒−푡푑푡 0

iii. Relationship between the Beta function and the Gamma function There are two different methods for deriving the relationship between the beta function and the gamma function: i. The formula of the beta function can be derived as a definite integral whose integrand depends on two variables 푎 and 푏 by reversing the order of integration of a double integral as follows: ∞ ∞ Γ(푎) Γ(푏) = 푡푎−1푒−푡 푑푡 푢푏−1푒−푢 푑푢 ퟎ ퟎ

∞ ∞ = 푡푎−1푢푏−1푒−(푡+푢)푑푡푑푢 ퟎ ퟎ

Using the transformation 푢 = 푡푣 we obtain ∞ ∞ Γ 푎 Γ 푏 = 푡푎−1(푡푣)푏−1푒−(푡+푡푣)푡푑푡푑푣 ퟎ ퟎ ∞ ∞ = 푡푎−1푡푏 푣푏−1푒−(푡+푡푣)푑푡푑푣 ퟎ ퟎ ∞ ∞ = 푡푎+푏−1푣푏−1푒−푡(1+푣)푑푡푑푣 ퟎ ퟎ

Using the transformation 푤 = 푡 푣 + 1 again gives ∞ ∞ 푤 푎+푏−1 = 푣푏−1푒−푤 (푣 + 1)−1푑푤푑푣 푣 + 1 ퟎ ퟎ

∞ ∞ 푤푎+푏−1푒−푤 = 푣푏−1푑푣 푑푤 (푣 + 1)푎+푏 ퟎ ퟎ ∞ ∞ 푣푏−1 = 푑푣 푤푎+푏−1푒−푤 푑푤 (푣 + 1)푎+푏 ퟎ ퟎ ∞ 푣푏−1 = 푑푣 Γ(a + b) 푣 + 1 푎+푏 ퟎ From which it follows that ∞ Γ(푎) Γ(푏) 푣푏−1 퐵 푎, 푏 = = 푑푣 Γ(푎 + 푏) 푣 + 1 푎+푏 ퟎ 푥 Finally, using the transformation 푣 = we get the alternative formulation 1 − 푥

∞ 푥 푏−1 1 퐵 푎, 푏 = 1 − 푥 푑푥 푥 푎+푏 (1 − 푥)2 ퟎ 1 − 푥 + 1

∞ 푥 ( )푏−1 1 = 1 − 푥 푑푥 1 푎+푏 (1 − 푥)2 ퟎ 1 − 푥

∞ = 푥푏−1(1 − 푥)−푏+1+푎+푏−2 푑푥 ퟎ

1 = 푥푏−1(1 − 푥)푎−1 푑푥 ퟎ which is the beta function. ii. Another method for deriving the relationship between beta function and gamma function is by using the polar co-ordinates as shown below.

∞ ∞ Γ(푎) Γ(푏) = 푥푎−1푒−푥 푑푥 푦푏−1푒−푦 푑푦 ퟎ ퟎ ∞ ∞ = 푒−(푥+푦)푥푎−1 푦푏−1푑푥푑푦 ퟎ ퟎ let 푥 = s2 and 푦 = t2 therefore d푥 = 2푠푑푠 and 푑푦 = 2푡푑푡 now ∞ ∞ 2 2 Γ(a)Γ(푏) = 푒−(푠 +t )s2(a−1) 푡2(푏−1). 2푠푑푠2푡푑푡 ퟎ ퟎ ∞ ∞ 2 2 = 4 푒−(푠 +t )s2a−1 푡2푏−1푑푠 푑푡 ퟎ ퟎ Next, let 푠 = 푟푠푖푛휃 푎푛푑 푡 = 푟푐표푠휃 Therefore 푑푠 = 푟푐표푠휃 푑휃 푎푛푑 푑푡 = −푟푠푖푛휃푑휃

푑푠 푑푡 푟퐶표푠휃 −푟푆푖푛휃 퐽 = 푑휃 푑휃 = = 푟퐶표푠2휃 + 푟푆푖푛2휃 = 푟 푑푠 푑푡 푆푖푛휃 퐶표푠휃 푑푟 푑푟

∞ 휋/2 2 ∴ 훤(푎)훤(푏) = 4 푒−푟 (푟푆푖푛휃)2푎−1(푟퐶표푠휃)2푏−1 퐽 푑휃푑푟 ퟎ ퟎ ∞ 휋/2 2 = 4 푒−푟 푟2푎 −1+2푏−1(푆푖푛휃)2푎−1(퐶표푠휃)2푏−1 푟푑휃푑푟 ퟎ ퟎ ∞ 휋/2 2 = 2 2 (푆푖푛휃)2푎−1 퐶표푠휃 2푏−1푑휃 푒−푟 푟2(푎+푏−1) 푟푑푟 ퟎ ퟎ

1 푎−1 푏−1 Recall that 퐵(푎, 푏) = 0 푥 (1 − 푥) 푑푥 Let 푥 = 푆푖푛2휃 ⟹ 푑푥 = 2푆푖푛휃퐶표푠휃푑휃 휋 2 ∴ 퐵 푎, 푏 = 푆푖푛휃 2푎−2 퐶표푠휃 2푏−2. 2푆푖푛휃퐶표푠휃푑휃 ퟎ 휋 2 = 2 푆푖푛휃 2푎−1 퐶표푠휃 2푏−1푑휃 ퟎ ∞ 2 ∴ Γ a Γ(b) = 2 퐵 푎, 푏 푒−푟 푟2(푎+푏−1) 푟푑푟 ퟎ ∞ 2 = 퐵 푎, 푏 푒−푟 푟2(푎+푏−1)2푟 푑푟 ퟎ Let 푢 = 푟2 ⟹ 푑푢 = 2푟푑푟 ∞ ∴ Γ a Γ(b) = 퐵 푎, 푏 푒−푢 푢푎+푏−1 푑푢 ퟎ = 퐵 푎, 푏 Γ(푎 + 푏) Γ a Γ b ∴ 퐵 푎, 푏 = Γ(푎 + 푏)

2.2 Construction of the Classical Beta Distribution

2.2.1 From the beta function

By definition

ퟏ 퐵 푎, 푏 = 푥푎−1(1 − 푥)푏−1푑푥 , 푎 > 0, 푏 > 0 ퟎ

Normalizing the beta function gives 1 푥푎−1(1 − 푥)푏−1푑푥 1 = 퐵 푎, 푏 0

Thus the probability density function of the beta distribution with shape parameters a and b is expressed as

푥푎−1(1 − 푥)푏−1 푓 푥; 푎, 푏 = , 0 < 푥 < 1, 푎 > 0, 푏 > 0 ______(2.1) 퐵(푎, 푏)

The cumulative distribution function of equation 2.1 is given by

1 푥 퐹 푥 = 푡푎−1(1 − 푡)푏−1 푑푡, 0 < 푡 < 1, 푎 > 0, 푏 > 0______(2.2) 퐵(푎, 푏) 0

Where the parameters 푎 and 푏 are positive real quantities and the variable x satisfies 0 ≤ 푥 ≤ 1.The quantity 퐵(푎, 푏) is the beta function. Equation (2.1) is also known as the standard beta or classical beta distribution.

2.2.2 From stochastic processes

Consider a Poisson process with arrival rate of 휆 events per unit time. Let 푤푘 denote the th th waiting time until the k arrival of an event and 푤푠 denote the waiting time until the s arrival, 푠 > 푘. then, 푤푘 and 푤푠 − 푤푘 are independent gamma random variables with 푤 ~gamma (푘, 1) and 푤 − 푤 ~ gamma (s − k, 1/λ). 푘 휆 푠 푘 The proportion of the time taken by the first k arrivals in the time needed for the first arrival is 푤 푤 푘 = 푘 ~ 푏푒푡푎 푘, 푠 − 푘 푤푠 푤푘+(푤푠−푤푘 ) In simple terms, a continuous beta distribution 퐵(푎, 푏) can be derived from a Poisson process such that if 푎 + 푏 events occur in a time interval, then the fraction of that interval until the 푎푡푕 event occurs has a 퐵(푎, 푏) distribution.

2.2.3 From ratio transformation of two independent gamma variables

Let 푋1 and 푋2 be independent gamma variables with joint pdf

1 푕 푥 , 푥 = 푥 훼−1푥 훽−1푒−푥1−푥2 , 0 < 푥 < ∞, 0 < 푥 < ∞ 1 2 Γ α Γ(β) 1 2 1 2 where 훼 > 0, 훽 > 0

푋1 Let 푌1 = 푋1 + 푋2 and 푌2= 푋1+푋2

푦1 = 푔1 (푥1, 푥2) = 푥1 + 푥2 푦2 = 푔2 (푥1, 푥2) =푥1/(푥1 + 푥2)

푥1 = 푕1(푦1, 푦2)= 푦1푦2 푥2 = 푕2(푦1, 푦2)= 푦1(1 − 푦2)

푦1 푦2 퐽 = = −푦1 1 – 푦2 −푦1

The transformation is one-to-one and maps 𝒜, the first quadrant of the 푥1푥2 plane onto

ℬ = {(푦1, 푦2): 0 < 푦1 < 1, 0 < 푦2 < 1}.

The joint pdf of 푌1, 푌2 is

훼−1 훽−1 −푦1 푓(푦1, 푦2) = 푦1/Γ(α)Γ(β) 푦1푦2 [푦1 1 − 푦2 ] 푒 훼−1 훽−1 푦2 1 − 푦2 = 푦 훼+훽 −1푒−푦1 , (푦 푦 ) ∈ ℬ. Γ(α)Γ(β) 1 1, 2

Because ℬ is a rectangular region and because g (푦1,푦2) can be factored into a function of 푦1 and a function of 푦2, it follows that 푌1 and 푌2 are statistically independent.

The marginal pdf of 푌2 is 훼−1 훽−1 ∞ 푦2 1 − 푦2 훼+훽 −1 −푦1 푓푌2 (푦2) = 푦1 푒 푑푦1 Γ(α)Γ(β) 0 Γ α + β = 푦 훼−1 1 − 푦 훽 −1______(2.3) Γ(α)Γ(β) 2 2 for 0 < 푦2 < 1.

This is the pdf of a beta distribution with parameters α and β.

Also, since 푓(푦1, 푦2) = 푓푌1 (푦1) 푓푌2 (푦2) it implies that 1 푓 (푦 ) = 푦 훼+훽−1푒−푦1 푌1 1 Γ α + β 1 for 0 < 푦1 < ∞.

Thus, 푌1 has a gamma distribution with parameter values α + β and 1.

2.2.4 From the rth order statistic of the Uniform distribution

Let 푋1, … , 푋푛 be independent and identically distributed random sample from the th standard Uniform distribution 푢(0,1). Let 푋푟 be the r order statistics from this sample.

Then the probability distribution of 푋푟 is a beta distribution with parameters r and n-r+1.

2.3 Properties of the Classical Beta Distribution

2.3.1 rth-order moments To obtain the rth-order moments of the classical beta distribution, we evaluate 1 E Xr = 푥푟 푓 푥; 푎, 푏 푑푥 0

1 푥푎−1(1 − 푥)푏−1 = 푥푟 푑푥 퐵(푎, 푏) 0

1 1 = 푥푎+푟−1(1 − 푥)푏−1 푑푥 B(푝, 푞) 0

1 1 푥(푎+푟)−1(1 − 푥)푏−1 = . 퐵(푎 + 푟, 푏)푑푥 퐵(푎, 푏) 퐵(푎 + 푟, 푏) 0

퐵(푎 + 푟, 푏) = 퐵(푎, 푏)

Γ(a + r) Γ(b) Γ(a + b) = . Γ(a + b + r) Γ(a)Γ(b)

Γ(a + r) Γ(a + b) = ______(2.4) Γ(a + b + r) Γ(a)

2.3.2 The first four moments

The mean is found by substituting r = 1 in equation (2.4)

Γ(a + 1) Γ(a + b) i. e., 푚 ≡ μ = E 푋 = 1 Γ(a + b + 1)Γ(a)

aΓ(a) Γ(a + b) = (a + b)Γ(a + b) Γ(a) a = (퐶푕푟푖푠푡푖푎푛 푊푎푙푐푘, 1996)______(2.5) a + b

Γ(a + b) Γ(a + 2) 푚 ≡ 퐸 푋2 = 2 Γ a + b + 2 Γ(a)

Γ(a + b) (a + 1)Γ(a + 1) = (a + b + 1)Γ(a + b + 1) Γ(a)

Γ a + b (a + 1) aΓ(a = a + b + 1 (a + b)Γ(a + b) Γ(a)

(a + 1) a = a + b + 1 (a + b)

a(a + 1) = (a + b) a + b + 1

(a + 1) = 푚 (퐶푎푡푕푒푟푖푛푒 푒푡. 푎푙. , 2010)______(2.6) 1 a + b + 1

Γ(a + b) Γ(a + 3) 푚 ≡ 퐸 푋3 = 3 Γ(a + b + 3) Γ(a)

Γ(a + b) a + 2 a + 1 aΓ(a) = a + b a + b + 2 a + b + 1 Γ(a + b) Γ(a)

(a + 2)(a + 1)a = a + b + 2 a + b + 1 (a + b)

a + 2 = 푚 (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010)______(2.7) 2 a + b + 2

Γ(a + b) Γ(a + 4) 푚 ≡ 퐸 푋4 = 4 Γ(a + b + 4) Γ(a)

Γ a + b a + 3 (a + 2) a + 1 aΓ(a) = a + b (a + b + 1) a + b + 2 a + b + 3 Γ(a + b) Γ(a)

(a + 3)(a + 2)(a + 1)a = (a + b + 3) a + b + 2 a + b + 1 (a + b)

a(a + 1) (a + 2)(a + 3) = (a + b) a + b + 1 a + b + 2 a + b + 3

(a + 3) = 푚 3 a + b + 3

2.3.3 Mode Mode is obtained by solving d 푓(푥; 푎, 푏) = 0 dx

d 푦푎−1 1 − 푥 푏−1 = 0 dx 퐵 푎, 푏

(푎 − 1)푥푎−2 1 − 푥 푏−1 + 푏 − 1 (−1) 1 − 푥 푏−2푥푎−1 = 0 퐵 푎, 푏

(푎 − 1)푥푎−2 1 − 푥 푏−1 = 푏 − 1 1 − 푥 푏−2푥푎−1

(푎 − 1)푥푎−1푥−1 1 − 푥 푏−1 = 푏 − 1 1 − 푥 푏−1(1 − 푥)−1푥푎−1

(푎 − 1)푥−1 = 푏 − 1 (1 − 푥)−1

푎 − 1 푥 = , 푎 > 1, 푏 > 1 푏 − 1 1 − 푥

1 − 푥 푎 − 1 = 푥(푏 − 1)

푎 − 1 − 푥푎 + 푥 = 푥푏 − 푥

푥 −푎 + 1 − 푏 + 1 = 1 − 푎

(푎 − 1) 푥 = , 푥 = 0 푖푓 푎 = 1 푎푛푑 푥 = 1 푖푓 푏 = 1______(2.8) (푎 + 푏 − 2)

2.3.4 Variance Since variance can be expressed as

Var X = 퐸 푋2 − 퐸 푋 2

2 = m2 − 푚1

We can now get the variance as

a(a + 1) a 2 Var X = 휍2 = − (a + b)(a + b + 1) a + b

a a + 1 (a + b)2 − a2 a + b (a + b + 1) = (a + b)(a + b + 1)(a + b)2

a a + 1 (a + b) − a2 a + b a + b + 1 = a + b + 1 a + b 2

a3 + a2b + a2 + ab − a3 − a2b + a2 = a + b + 1 a + b 2

ab = (퐶푕푟푖푠푡푖푎푛 푊푎푙푐푘, 1996)______(2.9) a + b + 1 a + b 2

2.3.5 Skewness

Skewness (the third central moment) can be expressed as

3 Skewness = 푢3 = 퐸(푋 − 퐸(푋)) = 퐸 푋3 − 3퐸 푋2 퐸 푋 + 2 퐸 푋 3

3 = m3 − 3푚2푚1+2푚1 ,

This can be manipulated algebraically as follows:

a a + 1 (a + 2) a2(a + 1) a 3 푢 = − 3 + 2 3 a + b a + b + 1 (a + b + 2) (a + b)2(a + b + 1) a + b

a a + 1 (a + 2) − a2 a + 1 (a + b + 2) 2a3 = + (a + b)2(a + b + 1)(a + b + 2) (a + b)3

a3 + 3a2 + 2a a + b − 3a4 − 3a3b − 9a3 − 3a2b − 6a2 2a3 = + a + b 2 a + b + 1 a + b + 2 (a + b)3

a4 + 3a3 + 2a2 + a3b + 3a2b + 2ab − 3a4 − 3a3b − 9a3 − 3a2b − 6a2 = a + b 2 a + b + 1 a + b + 2 2a3 + (a + b)3

−2a4 − 6a3 − 4a2 − 2a3b + 2ab 2a3 = + a + b 2 a + b + 1 a + b + 2 (a + b)3

(a + b)(−2a4 − 6a3 − 4a2 − 2a3b + 2ab) + (2a4 + 2a3b + 2a3)(a + b + 2) = a + b 3 a + b + 1 a + b + 2

−2a5 − 6a4 − 2a4b − 4a3 + 2a2b−2a4b − 6a3b − 2a3b2 − 4a2b + 2ab2+2a5+2a4b = a + b 3 a + b + 1 a + b + 2

+4a4 + 2a4b + 2a3b2 + 4a3b + 2a4 + 2a3b + 4a3

a + b 3 a + b + 1 a + b + 2

2ab2 − 2a2b = a + b 3 a + b + 1 a + b + 2

2ab(b − a) = (퐶푕푟푖푠푡푖푎푛 푊푎푙푐푘, 1996)______(2.10) a + b 3 a + b + 1 a + b + 2

The standardized skewness can be expressed as

푋 − 퐸 푋 3 퐸 휍

1 2ab b − a = 휍3 a + b 3 a + b + 1 a + b + 2 1 2ab b − a = 3 3 ab a + b a + b + 1 a + b + 2

a + b + 1 a + b 2

a + b + 1 a + b 2 a + b + 1 a + b 2ab b − a = . 푎푏 ab a + b 3 a + b + 1 a + b + 2 a + b + 1 2 b − a = ab a + b + 2

2 b − a a + b + 1 = ______(2.11) a + b + 2 ab

2.3.6 Kurtosis Kurtosis can be expressed as

4 3 2 2 4 Kurtosis = 푢4 = 퐸 푋 − 4퐸 푋 퐸 푋 + 6{퐸 푋 } 퐸 푋 − 3 퐸 푋

2 4 = m4 − 4푚1푚3+6푚1 푚2 − 3푚1 ,

a a + 1 a + 2 a + 3 = a + b a + b + 1 a + b + 2 a + b + 3 a a a + 1 a + 2 −4 a + b a + b a + b + 1 a + b + 2 a 2 a(a + 1) a 4 +6 − 3 a + b a + b (a + b + 1) a + b

a2 + a a2 + 5a + 6 = a + b a + b + 1 a + b + 2 a + b + 3 a2 a2 + 3a + 2 −4 (a + b)2 a + b + 1 a + b + 2

6a4 + 6a3 a 4 + − 3 (a + b)3(a + b + 1) a + b

a + b a4 + 5a3 + 6a2 + a3 + 5a2 + 6a − 4a4 + 12a3 + 8a2 (a + b + 3) = (a + b)2 a + b + 1 a + b + 2 a + b + 3 6 a + b (a4 + a3) − 3a4(a + b + 1) + (a + b)4(a + b + 1) a5 + 5a4 + 6a3 + a4 + 5a3 + 6a2 + a4b + 5a3b + 6a2b + a3b + 5a2b + 6ab − (4a5 + 4a4b + 12a4 + 12a4 + 12a3b + 36a3 + 8a3 + 8a2b + 24a2) = (a + b)2 a + b + 1 a + b + 2 a + b + 3 (6a5 + 6a4 + 6a4 + 6a3b) − 3a5 − 3a4b − 3a4) + (a + b)4(a + b + 1) −3a5 − 18a4 − 33a3 − 18a2 − 3a4b − 6a3b − 6a3b + 3a2b + 6ab = (a + b)2 a + b + 1 a + b + 2 a + b + 3 3a5 + 3a4 + 3a4q + 6a3b + (a + b)4(a + b + 1)

a2 + 2ab + b2 −3a5 − 18a4 − 33a3 − 18a2 − 3a4b − 6a3b − 6a3b + 3a2b + 6ab = a + b 4 a + b + 1 a + b + 2 a + b + 3 a2 + b2 + 2ab + 5a + 5b + 6 3a5 + 3a4 + 3a4b + 6a3b + a + b 4 a + b + 1 a + b + 2 a + b + 3 −3a7 − 18a6 − 33a5 − 18a4 − 3a6b − 6a5b + 3a4b − 6a6b + 36a5b − 66a4b −36a3b − 6a5b2 = a + b 4 a + b + 1 a + b + 2 a + b + 3

−12a4b2 + 6a3b2 + 12a2b2 − 3a5b2 − 18a4b2 − 33a3a2 − 18a2b2 − 3a4b3 3 3 2 3 + −6a b + 3a b a + b 4 a + b + 1 a + b + 2 a + b + 3 6ab3 + 3a7 + 3a6 + 3a6b + 6a5b + 3a5b2 + 3a4b2 + 3a4b3 + 6a3b3 + 6a6b 5 5 2 + +6a b + 6a b a + b 4 a + b + 1 a + b + 2 a + b + 3

12a4b2 + 15a6 + 15a5 + 15a5b + 30a4b + 15a5b + 15a4b2 + 30a3b2 + 18a5 4 4 3 + +18a + 18a b + 36a b a + b 4 a + b + 1 a + b + 2 a + b + 3

6a3b + 3a3b2 − 6a2b2 + 3a2b3 + 6ab3 = a + b 4 a + b + 1 a + b + 2 a + b + 3

3푎푏[2a2 + a2b − 2ab + ab2 + 2b2] = a + b 4 a + b + 1 a + b + 2 a + b + 3

3푎푏[a2(b + 2) − 2ab + b2(a + 2)] = (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010)______(2.12) a + b 4 a + b + 1 a + b + 2 a + b + 3 The standardized kurtosis can be expressed as 푋 − 퐸 푋 4 퐸 휍 1 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] = 휍4 a + b 4 a + b + 1 a + b + 2 a + b + 3 1 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] = ab 2 a + b 4 a + b + 1 a + b + 2 a + b + 3

a + b + 1 a + b 2 a + b + 1 2 a + b 4 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] = (푎푏)2 a + b 4 a + b + 1 a + b + 2 a + b + 3 a + b + 1 3[a2(b + 2) − 2ab + b2(a + 2)] = ______(2.13) 푎푏 a + b + 2 a + b + 3

The standardized skewness and kurtosis are often denoted by 훽1, 훽2 in literature.

2.4 Shapes of Classical Beta Distribution The shape of the distribution is governed by the two parameters a and b, and can be summarized as follows; i. When a=b, the distribution is symmetrical about ½, the case of arcsine distribution, uniform distribution and parabolic distribution, and the skewness is zero, as a and b become larger, the distribution becomes more peaked and the variance decreases

ii. When 푎 ≠ 푏, the distribution is skewed with the degree of skewness increasing as 푎

and 푏 become more unequal. In particular, if 푏 > 푎, then the skewness, 푢3 > 0 and the distribution is skewed to the right as in figure 2.2 and if 푎 > 푏, then the distribution is skewed to the left as in figure 2.1. iii. When a=b=1, the distribution becomes uniform distribution and the density function has a constant value 1 over the interval (0,1) for the random variable X. iv. Figures 2.2 and 2.3 below shows two cases of the beta unimodal distributions with 푎 > 0 and 푏 > 0. If a and/or b is less than one i.e 푎 − 1 푏 − 1 < 1 푓 0 → ∞ and /or f(1) → ∞ and the distribution is J- shaped. Figure 2.4 and 2.5 below show the beta distribution of two cases a=10, b=1 and a=1, b=10 respectively. v. The Classical beta distribution is U shaped if a < 1, and b < 1 as shown in figure 2.1 below.

Beta(0.66,0.5) 6

0 1

0.4 0.2 0.6 0.8

0.76 0.04 0.08 0.12 0.16 0.24 0.28 0.32 0.36 0.44 0.48 0.52 0.56 0.64 0.68 0.72 0.84 0.88 0.92 0.96

Fig.2. 1 : Left Skewedness

Beta(2,30) 14

0.7

0.98 0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 0.77 0.84 0.91

0.315 0.665 0.035 0.105 0.175 0.245 0.385 0.455 0.525 0.595 0.735 0.805 0.875 0.945

Fig.2. 2: Case of beta unimodal distribution with 푎 > 0

Beta(30,2) 14

0.7

0.98 0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 0.77 0.84 0.91

0.315 0.665 0.035 0.105 0.175 0.245 0.385 0.455 0.525 0.595 0.735 0.805 0.875 0.945

Fig.2. 3: Case of beta unimodal distribution with 푏 > 0

Beta(10,1) 12

0 1

0.8 0.2 0.4 0.6

0.84 0.88 0.92 0.96 0.04 0.08 0.12 0.16 0.24 0.28 0.32 0.36 0.44 0.48 0.52 0.56 0.64 0.68 0.72 0.76

Fig.2. 4: j-Shaped beta distribution

Beta(1,10) 12

0 1

0.6 0.2 0.4 0.8

0.56 0.64 0.04 0.08 0.12 0.16 0.24 0.28 0.32 0.36 0.44 0.48 0.52 0.68 0.72 0.76 0.84 0.88 0.92 0.96

Fig.2. 5: inverted j-Shaped beta distribution

2.5 Applications of the Classical Beta Distribution

Income and Wealth

Thurow (1970) applied the Classical beta distribution to US Census Bureau data of income distribution for households (families and unrelated individuals) for every year from 1949 – 1966, stratified by race. He also studied the impact of various macroeconomic factors on the parameters of the distribution via regression techniques. McDonald and Ransom (1979a) employed the Classical beta distribution for approximating US family incomes for 1960 and 1969 through 1975. When utilizing three different estimators, it turns out that the distribution is preferable to the gamma and lognormal distributions but inferior to Singh Maddala. McDonald (1984) estimated beta distribution of the first kind for 1970, 1975 and 1980 US family incomes.

2.6 Special Cases of the Classical Beta Distribution

2.6.1 Power Distribution Function This distribution is obtained when 푏 = 1 in the formula of the classical beta distribution. Thus, from equation (2.1) the pdf becomes

푥푎−1 푓 푥; 푎 = 퐵(푎, 1)

푥푎−1 = . Γ(푎 + 1) Γ(푎)Γ(1)

푥푎−1 = . 푎Γ(푎) Γ(푎)Γ(1)

= 푎푥푎−1, 푎 > 0 ______(2.14) and the CDF is given by

∞ 퐹 푥 = 푓(푥) 푑푥 0

∞ = 푎푥푎−1 푑푥 0

푥푎−1+1 = 푎 푎

= 푥푎 , 0 < 푎 < 1

Equation (2.14) is the power distribution. Generally letting 푥 = 푦/훼, with b=1 in equation (2.1) yield the general power distribution given by

푎푦푎−1 푓 푦 = , 0 < 푎 < 훼 훼푎 (Leemis and McQueston, 2008)

2.6.2 Properties of power distribution

i. rth-order moments

Putting b=1 in equation (2.4) Γ 푎 + 푟 Γ 푎 + 1 퐸 푋푟 = Γ a + 1 + r Γ a Γ(푎 + 푟)푎Γ(푎) = (a + r)Γ a + r Γ(a) 푎 = a + r

ii. Mean

푎 퐸 푋 = a + 1

iii. Mode 푎 − 1 푀표푑푒 = a + 1 − 2 푎 − 1 = = 1 a − 1

iv. Variance

푎 푉푎푟 푋 = (a + 2)(a + 1)2

v. Skewness

2푎(1 − 푎) 푆푘푒푤푛푒푠푠 = (a + 1)3(a + 2)(a + 3) While the standardized skewness is

2푎(1 − 푎) (a + 2)

(a + 3) a

vi. Kurtosis 3푎[푎2 1 + 2 − 2푎 + 푎 + 2 ] 퐾푢푟푡표푠푖푠 = a + 1 4 a + 1 + 1 a + 1 + 2 (a + 1 + 3) 3푎[3푎2 − 푎 + 2] = a + 1 4 a + 2 a + 3 (a + 4) and the standardized kurtosis is expressed as 푎 + 2 3 3푎2 − 푎 + 2

푎 a + 3 a + 4

2.6.3 Shape of Power distribution

The figure 2.6 below shows a graph of a power distribution with the parameter 푎 = 0.05. This is an example of Power distribution graph, being used to demonstrate ranking of popularity. To the right is the long tail, and to the left are the few that dominate. Varying 푎 gives different shapes of the power distribution. Special cases of 푎 = 1 (Uniform distribution), 푎 =2 (triangular distribution), 푎 < 1 gives inverted J-shaped distributions and 푎 > 2 (J-shaped distributions).

9 8 Power Distribution 7 6 5 4 3 2 1

0 1

0.2 0.4 0.6 0.8

0.04 0.08 0.12 0.16 0.24 0.28 0.32 0.36 0.44 0.48 0.52 0.56 0.64 0.68 0.72 0.76 0.84 0.88 0.92 0.96

Fig.2. 6: Power distribution

2.6.4 Uniform Distribution Function

Special case when 푎=b=1 in equation (2.1) 푓 푥 = 1 ______(2.15) Equation (2.15) is a Uniform distribution on the interval [0,1]. The Uniform distribution function is a special case of the power distribution function when a=1.

2.6.5 Properties of the Uniform distribution

i. rth-order moments Putting 푎 = 1 and 푏 = 1 in equation 2.4 Γ 1 + 푟 Γ 1 + 1 퐸 푥푟 = Γ 1 + 1 + r Γ 1 rΓ 푟 . 1 = 1 + r r Γ r . 1 1 = 1 + r

ii. Mean 푎 mean = , 푎 + 푏 Letting a=b=1, 1 = = 0.5 1 + 1

iii. Mode

Mode is any value in the interval [0,1]

iv. Variance From the variance of the standard beta distribution given by

푎푏 Variance = (푎 + 푏 + 1)(푎 + 푏)2 a=b=1, 1 = 3 2 2 1 = = 0.0833 12

v. Skewness

2푎푏(b − a) Skewness = = 0 a + b 3 a + b + 1 a + b + 2

2(1 − 1) (1 + 2) Standardized skewness = = 0 (1 + 3) 1

vi. Kurtosis

3푎푏[a2(b + 2) − 2ab + b2(a + 2)] kurtosis = a + b 4 a + b + 1 a + b + 2 a + b + 3

3 3 − 2 + 3 = 2 4 3 4 5 1 = = 0.0125 16 5 The standardized kurtosis is given by

푎 + 2 3 3푎2 − 푎 + 2

푎 a + 3 a + 4

1 + 2 3 3 − 1 + 2 = 1 1 + 3 1 + 4

3 4 = 3 = 1.8 4 5

2.6.6 Shape of Uniform distribution Figure 2.7 below shows a graphical representation of the Uniform distribution on the interval (0,1)

Beta(1,1) 1.2

0.8

0.6

0.4

0.2

0.9

0.27 0.09 0.18 0.36 0.45 0.54 0.63 0.72 0.81 0.99

0.675 0.045 0.135 0.225 0.315 0.405 0.495 0.585 0.765 0.855 0.945

Fig.2. 7: Uniform distribution on the interval (0,1)

2.6.7 Arcsine Distribution Function

Special case of the standard beta distribution is the arcsine distribution, when a=b=1/2 in equation (2.1). In this special case, we have 1 1 푥1/2−1 1 − x 1/2−1 푓 푥; , = , 0 < 푥 < 1 2 2 1 1 퐵 2 , 2

푥−1/2 1 − x −1/2 푓 푥 = , 0 < 푥 < 1 1 1 퐵 2 , 2 1 푓 푥 = , 0 < 푥 < 1 ______(2.16) 휋 푥(1 − 푥) and the cumulative distribution function is given by 2 F x = arcsin 푥 . 휋 Note: Γ(1/2) = 휋 Equation (2.16) is an Arcsine distribution

2.6.8 Properties of the arcsine distribution

i. rth-order moments Putting 푎 = 1 and 푏 = 1 in equation 2.4 2 2 1 Γ + 푟 Γ 1 퐸 푋푟 = 2 Γ 1 + r Γ 1 1 Γ + 푟 = 2 1

rΓ r Γ 2

ii. Mean From the mean of the beta distribution given by

푎 mean = , 푎 + 푏 1 a = b = 2 1 1 = 2 = = 0.5 1 1 2 2 + 2

iii. Mode 1 푎 − 1 − 1 mode = = 2 푎 + 푏 − 2 1 1 2 + 2 − 2 1 − 1 = 2 1 1 2 + 2 − 2 1 = 2

iv. Variance From the variance of the beta distribution given by

푎푏 Variance = (푎 + 푏 + 1)(푎 + 푏)2 1 a = b = 2 1 = 4 = 0.125 2 1 2

v. Skewness 2푎푏(b − a) Skewness = = 0 a + b 3 a + b + 1 a + b + 2

2(b − a) 푎 + 푏 + 1 Standardized skewness = a + b + 2 푎푏 1 1 2( − ) 1 + 1 = 2 2 = 0 1 + 2 1 4

vi. Kurtosis 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] kurtosis = a + b 4 a + b + 1 a + b + 2 a + b + 3

1 1 1 1 1 1 3 [ ( + 2) − 2( ) + ( + 2)] = 4 4 2 4 4 2 1 4 1 + 1 1 + 2 1 + 3

3 5 4 5 [( ) − ( ) + ( )] = 4 8 8 8 2 3 4

= 32 = 0.0234 24

1 1 1 1 1 1 + 1 3 + 2 − 2 + + 2 Standardized kurtosis = 4 2 4 4 2 = 1.5 1 1 + 2 1 + 3 4

2.6.9 Shape of Arcsine distribution

Arcsine Distribution 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

0.325 0.365 0.405 0.005 0.045 0.085 0.125 0.165 0.205 0.245 0.285 0.445 0.485 0.525 0.565 0.605 0.645 0.685 0.725 0.765 0.805 0.845 0.885 0.925 0.965

Fig.2. 8: Arcsine distribution

2.6.10 Triangular shaped distributions (a=1, b=2)

For a=1 and b=2 we get triangular shaped distribution as follows 1 − x f x; 1,2 = B 1,2 = 2 − 2푥 ______(2.17) With the cdf given as

퐹 푥 = 2푥 − 푥2

2.6.11 Properties of the Triangular shaped distribution (a=1, b=2)

i. rth-order moments Putting 푎 = 1 and 푏 = 2 in equation 2.4 Γ 1 + 푟 Γ 3 퐸 푋푟 = Γ 3 + r Γ 1 2rΓ 푟 = Γ r + 3

ii. Mean From the mean of the standard beta distribution given by 푎 mean = , 푎 + 푏 a = 1, b = 2 1 = = 0.3333 3

iii. Mode 푎 − 1 1 − 1 mode = = = 0 푎 + 푏 − 2 1 + 2 − 2

iv. Variance From the variance of the standard beta distribution given by

푎푏 Variance = (푎 + 푏 + 1)(푎 + 푏)2 a = 1, b = 2

2 = 4 3 2 1 = = 0.0556 18 Standard deviation, 휍 = 푣푎푟 = 0.2358 휍 0.2358 퐶푉 = = = 0.7074 휇 0.3333

v. Skewness 2푎푏(b − a) Skewness = a + b 3 a + b + 1 a + b + 2 2(2)(1) = 3 3 4 5 1 = = 0.0074 3 3 5

2(b − a) 푎 + 푏 + 1 Standardized skewness = a + b + 2 푎푏 a = 1, b = 2 2 = 2 = 0.5657 5

vi. Kurtosis 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] kurtosis = a + b 4 a + b + 1 a + b + 2 a + b + 3 3 2 [1(4) − 4 + 4(3)] = 3 4 4 5 6 1 = = 0.0074 3 3 5 3 + 1 3 1 2 + 2 − 2 1 2 + 4 1 + 2 Standardized kurtosis = = 2.4 2 3 + 2 3 + 3

2.6.12 Shape of triangular distribution (a=1, b=2)

Beta(1,2) 2.5

1.5

0.5

0.9

0.36 0.09 0.18 0.27 0.45 0.54 0.63 0.72 0.81 0.99

0.405 0.765 0.045 0.135 0.225 0.315 0.495 0.585 0.675 0.855 0.945

Fig.2. 9: Triangular shaped distribution

2.6.13 Triangular shaped distributions (a=2, b=1) For a=2 and b=1 we get triangular shaped distribution as follows

푥 푓 푥; 2,1 = 퐵(2,1)

= 2푥______(2.18) The CDF of equation 2.18 is given by 퐹(푥) = 푥2

2.6.14 Properties of the Triangular shaped distribution (a=2, b=1)

i. rth - order moments Putting 푎 = 2 and 푏 = 1 in equation 2.4 Γ 2 + 푟 Γ 3 퐸 푋푟 = Γ 3 + r Γ 2

2Γ 2 + 푟 = Γ r + 3

ii. Mean From the mean of the beta distribution given by 푎 mean = , 푎 + 푏 a=2, b=1, 2 = = 0.6667 3

iii. Mode 푎 − 1 2 − 1 mode = = = 1 푎 + 푏 − 2 2 + 1 − 2

iv. Variance From the variance of the beta distribution given by

푎푏 Variance = (푎 + 푏 + 1)(푎 + 푏)2 a=2, b=1, 2 = 4 3 2 1 = = 0.0556 18

v. Skewness 2푎푏(b − a) Skewness = a + b 3 a + b + 1 a + b + 2 2(2)(−1) = 3 3 4 5 −1 = = −0.0074 3 3 5

2(b − a) 푎 + 푏 + 1 Standardized skewness = a + b + 2 푎푏 a=2, b=1 −2 = 2 = −0.5657 5

vi. Kurtosis 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] kurtosis = a + b 4 a + b + 1 a + b + 2 a + b + 3 3 2 [4(3) − 4 + 1(4)] = 3 4 4 5 6

1 = = 0.0074 3 3 5 3 + 1 3 4 1 + 2 − 2 2 1 + 1 2 + 2 Standardized kurtosis = = 2.4 2 3 + 2 3 + 3

2.6.15 Shape of triangular distribution (a=2, b=1)

Beta(2,1) 2.5

1.5

0.5

0.4

0.31 0.04 0.13 0.22 0.49 0.58 0.67 0.76 0.85 0.94

0.355 0.085 0.175 0.265 0.445 0.535 0.625 0.715 0.805 0.895 0.985

Fig.2. 10: Trangular shaped distribution

2.6.16 Parabolic shaped distribution

For a=b=2 we obtain a distribution of parabolic shape, 푥(1 − 푥) 푓 푥; 2,2 = 퐵(2,2)

= 6푥(1 − 푥)______(2.19)

The cdf of equation 2.19 is given by

퐹(푥) = 3푥2 − 2푥3

2.6.17 Properties of parabolic shaped distribution

i. rth-order moments Putting 푎 = 2 and 푏 = 2 in equation 2.4 Γ 2 + 푟 Γ 4 퐸 푋푟 = Γ 4 + r Γ 2 6 = r + 3 (r + 2)

ii. Mean From the mean of the Classical beta distribution given by 푎 E(X) = , 푎 + 푏 a = b = 2,

2 = = 0.5 2 + 2

iii. Mode From the mode of the Classical beta distribution given by

푎 − 1 Mode = (푎 + 푏 − 2) a = b = 2, = 0.5

iv. Variance From the variance of Classical beta distribution given by

푎푏 Variance = (푎 + 푏 + 1)(푎 + 푏)2 a = b = 2, 4 = 5 4 2 1 = = 0.05 20

standard deviation, 휍 = 0.05 = 0.2236

0.2236 퐶푉 = = 0.4472 0.5

v. Skewness From the skewness of the Classical beta distribution given by

2푎푏(푏 − 푎) Skewness = 푎 + 푏 푎 + 푏 + 1 (푎 + 푏 + 2) a=b=2, = 0

2(b − a) 푎 + 푏 + 1 Standardized skewness = = 0 a + b + 2 푎푏

vi. Kurtosis 3푎푏[a2(b + 2) − 2ab + b2(a + 2)] kurtosis = a + b 4 a + b + 1 a + b + 2 a + b + 3 3 4 [4(4) − 8 + 4(4)] = 4 4 5 6 7

3 = = 0.0054 4 2 5 7

4 + 1 3 4 2 + 2 − 2 2 2 + 4 2 + 2 Standardized kurtosis = = 2.1429 4 4 + 2 4 + 3

2.6.18 Shape of Parabolic distribution The figure 2.11 below shows the shape of the parabolic distribution.

Beta(2,2) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

0.4

0.94 0.04 0.13 0.22 0.31 0.49 0.58 0.67 0.76 0.85

0.895 0.985 0.085 0.175 0.265 0.355 0.445 0.535 0.625 0.715 0.805

Fig.2. 11: Parabolic distribution

2.7 Transformation of Special Case of the Classical Beta Distribution

2.7.1 Wigner Semicircle Distribution Function

Letting a=b=3/2 in equation (2.1) we obtain the following distribution function 3 3 푥3/2−1 1 − 푥 3/2−1 푓 푥; , = , 0 < 푥 < 1 2 2 3 3 퐵 2 , 2

2. 푥1/2 1 − 푥 1/2 푓 푥 = 2 , 0 < 푥 < 1 1 2 휋

8 푥(1 − 푥) 푓 푥 = 휋

Let

푦 + 푅 1 푥 = , |퐽| = 2푅 2푅 Limits changes as 푥 = 0 → 푦 > −푅 and 푥 = 1 → 푦 = 푅

푦 + 푅 푦 + 푅 8 2푅 1 − 2푅 1 푓 푦 = . 휋 2푅 2 푦 + 푅 푦 + 푅 4 2푅 − 2푅 = 푅휋 2푅푦 + 2푅2 − 푦2 − 2푦푅 − 푅2 4 4푅2 = 푅휋 2 푅2 − 푦2 = , −푅 < 푦 < 푅 ______(2.20) 푅2휋 Equation (2.20) is the Wigner semicircle distribution.

2.7.2 Properties of Wigner Semicircle

i. rth-order moments The rth-order moments for positive integers r, the 2rthmoments are given by

푅 2푟 퐸 푌2푟 = 퐶 2 푟

Where 퐶 is the rth catalan number given by 퐶 = 1 2푟 so that the moments are the 푟 푟 푟+1 푟 catalan numbers if R=2. The odd numbers are zero.

ii. Mean Mean of Wigner semi-circle, the first order moment, odd moment is zero.

iii. Mode Mode of Wigner semi-circle is given by

푑 푓(푦) = 0 푑푦

2 푑 푅2 − 푦2 = 0 푅2휋 푑푦

푦 = 0

iv. Variance The variance of Wigner semi-circle is given by

Var Y = 퐸 푌2 − 퐸(푌)

푅 2 1 = . 2 2 2

푅2 = 4

v. Skewness The Skewness of Wigner semi-circle is zero, since it is the third order moment which is an odd moment.

vi. Kurtosis The Kurtosis of Wigner semi-circle is given by Kurtosis = 퐸 푌4 − 4퐸 푌 퐸 푌3 + 6 퐸 푌 2퐸 푌2 − 3 퐸 푌 4

= 퐸 푌4

푅 2.2 1 2.2 = 2 2 + 1 2

푅4 = 8

푅4 4 2 Standardized Kurtosis = . = 2 8 푅2

2.7.3 Shape of Wigner semicircle distribution The figure 2.12 below shows the shape of the Wigner semicircle distribution.

Beta(1.5,1.5) 1.4 1.2 1 0.8 0.6 0.4 0.2

0.4

0.22 0.85 0.04 0.13 0.31 0.49 0.58 0.67 0.76 0.94

0.175 0.085 0.265 0.355 0.445 0.535 0.625 0.715 0.805 0.895 0.985

Fig.2. 12: Wigner semicircle distribution

3 Chapter III: Type II Two Parameter Inverted Beta Distribution

3.1 Construction of Beta Distribution of the Second Kind

The beta distribution of the second kind is derived by using the transformation y 푇 : 푥 = 1 1 + y

in equation 2.1 to give a new distribution as follows:

푓(푦; 푎, 푏) = 푓(푥; 푎, 푏)|퐽| The limits changes as follows 푥 + 푥푦 = 푦 ⟹ 푦 1 − 푥 = 푥, 푦 = 푥 1−푥 when x = 0, y = 0 and when x = 1, y = ∞ ⟹ 0 < 푦 < ∞

Where

1 |퐽| = (1 + y)2

Therefore

푥푎−1 1 − 푥 푏−1 푓 푦; 푎, 푏 = . |퐽| 퐵 푎, 푏

푏−1 푦 푎−1 푦 (1 + 푦) 1 − 1 + 푦 1 = . 퐵 푎, 푏 (1 + y)2

푦푎−1 = , 0 < 푦 < ∞, 푎, 푏 > 0______(3.1) 퐵 푎, 푏 (1 + y)a+b

Equation (3.1) is a beta distribution of the second kind. It is also known as the standard form of a Pearson Type VI distribution, sometimes called beta prime distribution or Inverted beta distribution.

Another method of deriving beta distribution of the second kind is by using the transformation

1 푥 = in equation (2.1) 1 + y

푑푥 −1 = 푑푦 1 + y 2

The limits changes as follows 1 − x 푥 + 푥푦 = 1 ⟹ 푦 = , when x = 0, y = ∞ and when x = 1, y = 0 ⟹ 0 < 푦 < ∞ x 푓(푦; 푎, 푏) = 푓(푥; 푎, 푏)|퐽|

푏−1 1 푎−1 1 1 − 1 + 푦 1 + 푦 1 = . 퐵 푎, 푏 1 + y 2

푏−1 1 푎−1 푦 (1 + 푦) 1 + 푦 1 = . 퐵 푎, 푏 1 − y 2 푦푏−1 = 퐵 푎, 푏 1 + 푦 a+b

3.2 Properties of Beta Distribution of the Second Kind

3.2.1 rth-order moments To obtain rth-order moments of the beta distribution of the second kind, we evaluate

∞ 퐸 푌푟 = 푦푟 푓 푦; 푎, 푏 푑푦 0

∞ 푦푎−1 = 푦푟 푑푦 퐵(푎, 푏)(1 + y)a+b 0

∞ 1 푦푟+푎−1 = 푑푦 퐵(푎, 푏) (1 + y)a+b 0

To integrate the integral in above expression, substitute 푥 푦 = 1 − 푥 푑푦 1 = 푑푥 (1 − 푥)2

푦 1 − 푥 = 푥 ⟹ 1 + 푥 = 1 , and limits of integration become 0 to 1. Then (1−푦) substituting these terms in the above integral, we have

∞ 1 푦푟+푎−1 푥 푑푦 = ( )푟+푎−1 1 − 푥 푎+푏 (1 − 푥)−2 푑푥 (1 + y)a+b 1 − 푥 0 0

1 = 푥푟+푎−1(1 − 푥)−푟−푎+1+푎+푏−2 푑푥 0

1 = 푥(푟+푎)−1(1 − 푥)(푏−푟)−1 푑푥 0

= 퐵 푎 + 푟, 푏 − 푟 , 푟 < 푏

Consequently, the moments of beta distribution of the second kind are given by 퐵 푎 + 푟, 푏 − 푟 퐸 푌푟 = 퐵 푎, 푏 Γ 푎 + 푟 Γ 푏 − 푟 = 푟 < 푏 ______(3.2) Γ 푎 Γ 푏 Using recursive relationship Γ a + 1 = aΓ(a), the first four moments about zero can be written as follows

3.2.2 Mean

The mean can be obtained by substituting r=1 in equation 3.2

퐵 푎 + 푟, 푏 − 푟 푚 ≡ μ = E Y = 1 퐵 푎, 푏

Γ a + 1 Γ(b − 1) Γ a + b = . Γ(a + b) Γ(a)Γ(b)

aΓ a Γ b − 1 Γ a + b = . Γ a + b Γ(a)Γ(b)

aΓ b − 1 = Γ(b)

a b − 2 ! = 푏 − 1 !

푎 푏 − 2 ! = 푏 − 1 푏 − 2 ! 푎 = , 푏 > 1(퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010)______(3.3) 푏 − 1

If b ≤ 1, the mean is infinite i. e undefined

퐵 푎 + 2, 푏 − 2 푚 ≡ E Y2 = 2 퐵 푎, 푏

Γ a + 2 Γ(b − 2) Γ a + b = . Γ(a + b) Γ(a)Γ(b)

(a + 1)aΓ a Γ b − 2 = Γ(a)Γ(b)

a(a + 1) b − 3 ! = b − 1 b − 2 (b − 3)!

a(a + 1) = 푏 − 1 푏 − 2

(a + 1) = 푚 , 푏 > 2 (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010) 1 푏 − 2

퐵 푎 + 3, 푏 − 3 푚 ≡ E Y3 = 3 퐵 푎, 푏

Γ a + 3 Γ(b − 3) Γ a + b = . Γ(a + b) Γ(a)Γ(b)

a + 2 a + 1 aΓ a b − 4 ! = Γ(a) b − 1 b − 2 b − 3 (b − 4)!

a a + 1 (a + 2) = b − 1 b − 2 (b − 3)

푚 (a + 2) = 2 , 푏 > 3 (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010) (b − 3)

퐵 푎 + 4, 푏 − 4 푚 ≡ E Y4 = 4 퐵 푎, 푏

Γ a + 4 Γ(b − 4) Γ a + b = . Γ(a + b) Γ(a)Γ(b)

(a + 3)(a + 2)(a + 1)aΓ a b − 5 ! = Γ a b − 1 b − 2 b − 3 b − 4 (b − 5)!

a a + 1 a + 2 (a + 3) = b − 1 b − 2 b − 3 (b − 4)

푚 (a + 3) = 3 , 푏 > 4 (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010) (b − 4)

3.2.3 Mode Mode is obtained by solving

푑 푓 푦 = 0 푑푦 푑 푦푏−1 = = 0 푑푦 퐵 푎, 푏 1 + 푦 a+b

1 푏 − 1 푦푏−2. 1 + 푦 a+b − a + b 1 + 푦 a+b−1. 푦푏−1 = = 0 퐵 푎, 푏 1 + 푦 2a+2b

= 푏 − 1 푦푏−2. 1 + 푦 a+b − a + b 1 + 푦 a+b−1. 푦푏−1 = 0

푏 − 1 푦푏−2. 1 + 푦 a+b = a + b 1 + 푦 a+b−1. 푦푏−1

푏 − 1 푦푏−1. 푦−1. 1 + 푦 a+b = a + b 1 + 푦 a+b 1 + 푦 −1. 푦푏−1

푏 − 1 푦−1 = (a + b) 1 + 푦 −1

1 + 푦 푏 − 1 = 푦(a + b)

푏 − 1 + 푦푏 − 푦 = 푦푎 + yb

푦(푎 + 1) = 푏 − 1

푏 − 1 푦 = , if b ≥ 1, 0 otherwise______(3.4) 푎 + 1

3.2.4 Variance

Since variance is given by 퐸 푌2 − 퐸 푌 2, we have

a(a + 1) 푎2 푉푎푟(푌) = − 푏 − 1 푏 − 2 푏 − 1 2

푏 − 1 푎2 + 푎 − 푎2(푏 − 2) = 푏 − 1 2(푏 − 2)

푎2푏 + 푎푏 − 푎2 − 푎 − 푎2푏 + 2푎2 = 푏 − 1 2(푏 − 2)

푎푏 + 푎2 − 푎 = 푏 − 1 2(푏 − 2)

푎(푎 + 푏 − 1) = , 푏 > 2 (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010)______(3.5) 푏 − 1 2(푏 − 2)

3.2.5 Skewness

Skewness can be expressed as

3 2 3 Skewness = 푢3 = 퐸 푌 − 3퐸 푌 퐸 푌 + 2 퐸 푌

3 = m3 − 3푚2푚1+2푚1 ,

푎 푎 + 1 (푎 + 2) a(a + 1) 푎 푎 = − 3 + 2( )3 푏 − 1 푏 − 2 (푏 − 3) 푏 − 1 푏 − 2 푏 − 1 푏 − 1

푎2 + 푎 (푎 + 2) 3a3 + 3a2 2a3 = − + 푏 − 1 푏 − 2 (푏 − 3) 푏 − 1 2 푏 − 2 푏 − 1 3

푎3 + 3푎2 + 2푎 3a3 + 3a2 2a3 = − + 푏 − 1 푏 − 2 (푏 − 3) 푏 − 1 2 푏 − 2 푏 − 1 3

푏 − 1 2 푎3 + 3푎2 + 2푎 − 푏 − 1 푏 − 3 3a3 + 3a2 + 2a3 푏 − 2 (푏 − 3) = 푏 − 1 3 푏 − 2 (푏 − 3) (푏2 − 2푏 + 1) 푎3 + 3푎2 + 2푎 − (푏2 − 4푏 + 3) 3a3 + 3a2 + 2a3(푏2 − 5푏 + 6) = 푏 − 1 3 푏 − 2 (푏 − 3) 푎3푏2 + 3푎2푏2 + 2푎푏2 − 2푎3푏 − 6푎2푏 − 4푎푏 + 푎3 + 3푎2 + 2푎 − 3a3푏2 − 3a2푏2 + 12a3b +12푝2푏 − 9푎3 − 9푎2 + 2p3푏2 − 10a3푏 + 12a3 = 푏 − 1 3 푏 − 2 (푏 − 3)

2푎푏2 + 6푎2푏 − 4푎푏 + 4푎3 − 6푎2 + 2푎 = 푏 − 1 3 푏 − 2 푏 − 3

2푎[푏2 + 3푎푏 − 2푏 + 2푎2 − 3푎 + 1] = 푏 − 1 3 푏 − 2 (푏 − 3)

2푎[2푎2 + 3푎푏 − 3푎 + 푏2 − 2푏 + 1] = 푏 − 1 3 푏 − 2 (푏 − 3)

2푎[2푎2 + 3푎 푏 − 1 + (푏 − 1)2] = , 푏 > 3______(3.6) 푏 − 1 3 푏 − 2 (푏 − 3)

(퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010) Since skewness is positive, it can be concluded that the beta distribution of the second kind has usually along tail to the right.

3.2.6 Kurtosis

Kurtosis can be expressed as 4 3 2 2 4 Kurtosis = 푢4 = 퐸 푌 − 4퐸 푌 퐸 푌 + 6{퐸 푌 } 퐸 푌 − 3 퐸 푌

2 4 = m4 − 4푚1푚3+6푚1 푚2 − 3푚1 ,

a a + 1 a + 2 (a + 3) 푎 a a + 1 (a + 2) = − 4 b − 1 b − 2 b − 3 (b − 4) 푏 − 1 b − 1 b − 2 (b − 3) 푎 2 a(a + 1) 푎 + 6 − 3( )4 푏 − 1 푏 − 1 푏 − 2 푏 − 1

(a2 + a)(a2 + 5a + 6) 4(a3 + a2)(a + 2) 6(a4 + a3) = − + b − 1 b − 2 b − 3 (b − 4) b − 1 2 b − 2 (b − 3) b − 1 3 푏 − 2 3a4 − b − 1 4 (a4 + 5a3 + 6a2 + a3 + 5a2 + 6a) (4a4 + 8a3 + 4a3 + 8a2) (6a4 + 6a3) = − + b − 1 b − 2 b − 3 (b − 4) b − 1 2 b − 2 (b − 3) b − 1 3 푏 − 2 3a4 − b − 1 4 b − 1 3 a4 + 5a3 + 6a2 + a3 + 5a2 + 6a − b − 1 2(b − 4)(4a4 + 8a3 + 4a3 + 8a2) = b − 1 4 b − 2 b − 3 (b − 4)

b − 1 b − 3 b − 4 6a4 + 6a3 − b − 1 b − 2 b − 3 (b − 4)3a4 + b − 1 4 b − 2 b − 3 (b − 4)

(b − 1) b2 − 2b + 1 a4 + 6a3 + 11a2 + 6a − b2 − 2b + 1 (b − 4)(4a4 + 12a3 = b − 1 4 b − 2 b − 3 (b − 4) +8a2) b2 − 4b + 3 b − 4 6a4 + 6a3 − (b − 2)(b2 − 7b + 12)3a4

b − 1 4 b − 2 b − 3 b − 4 (b3 − 3b2 + 3b − 1) a4 + 6a3 + 11a2 + 6a − b3 − 6b2 + 9b − 4 (4a4 + 12a3 = b − 1 4 b − 2 b − 3 (b − 4) +8a2) b3 − 8b2 + 19b − 12 6a4 + 6a3 − (b − 2)(3a4b2 − 21a4b + 36a4)

b − 1 4 b − 2 b − 3 b − 4 = [(a4b3 + 6a3b3 + 11a2b3 + 6ab3)+(−3a4b2−18a3b2−33a2b2 − 18ab2)+(3a4b + 18a3b + 33a2b + 18ab ) + −a4 − 6a3 − 11a2 − 6a + −4a4b3 − 12a3b3 − 8a2b3 + (24a4b2 + 72a3b2 + 48a2b2)+(−36a4b − 108a3b − 72a2b) + 16a4 + 48a3 + 32a2+6a4b3−48a4b2+114a4b−72a4+6a3b3−48a3b2+114a3b−72a3+−3a4b3+27 a4b2−78a4b+72a4]/b−14b−2b−3b−4

3a2b3 + 6ab3+6a3b2+15a2b2 − 18ab2+3a4b + 24a3b − 39a2b + 18ab + 15a4 − 30a3 +21a2 − 6a = b − 1 4 b − 2 b − 3 b − 4 3a(ab3 + 2b3+2a2b2 + 5ab2 − 6b2+a3b + 8a2b − 13ab + 6b + 5a3 − 10a2 +7a − 2) = ______(3.7) b − 1 4 b − 2 b − 3 b − 4 (퐶푎푡푕푒푟푖푛푒 푒푡 푎푙. , 2010)

3.2.7 Shapes of Beta distribution of the second kind

12 B(0.5,0.5)

10 B(1,1) B(1,2) 8 B(2,1) 6 B(2,2) B(10,1) 4 B(1,10) 2 B(2,30)

0.55 0.11 0.22 0.33 0.44 0.66 0.77 0.88 0.99

0.825 0.055 0.165 0.275 0.385 0.495 0.605 0.715 0.935

Fig.3. 1: Beta distribution of the second kind

3.3 Application of Beta Distribution of the Second Kind

Vartia and Vartia (1980) fitted a four parameter shifted beta distribution of the second kind under the name of scaled and shifted F distribution to the 1967 distribution of taxed income in Finland. Using maximum likelihood and method of moment‘s estimators, they found that their model fits systematically better than the two- and three-parameter lognormal distributions. McDonald (1984) estimated the beta distribution of the second kind and both beta of the first and second kind distributions for 1970, 1975 and 1980 US family incomes.

3.4 Special Cases of Beta Distribution of the Second Kind

3.4.1 Lomax (Pareto II) distribution function Special case when a=1 in equation (3.1) gives the following density function 1 푓 푥; 푏 = , 0 < 푥 < ∞, 푏 > 0 퐵 1, 푏 1 + x 1+b

1 퐵 1, 푏 = 푏

푏 = , 푥 > 0 , 푏 > 0______(3.8) (1 + x)1+b Equation (3.8) is a Lomax distribution.

3.4.2 Properties of Lomax distribution

i. rth-order moments

Putting 푎 = 1 in equation 3.2 Γ 1 + 푟 Γ 푏 − 푟 퐸 푋푟 = Γ 1 Γ 푏 Γ 1 + 푟 Γ 푏 − 푟 = Γ 푏

ii. Mean Γ 1 + 1 Γ 푏 − 1 E(X) = Γ 푏 b − 2 ! = b − 1 b − 2 !

1 = 푏 − 1

iii. Mode 푏 − 1 Mode = 2

iv. Variance 푏 푉푎푟(푋) = (푏 − 2)(푏 − 1)2

v. Skewness 2 2 + 3 푏 − 1 + 푏 − 1 2 Skewness = 푏 − 1 3 푏 − 2 푏 − 3 2푏 푏 + 1 = 푏 − 1 3 푏 − 2 푏 − 3

vi. Kurtosis 3(3b3 + b2 + 2b) kurtosis = b − 1 4 b − 2 b − 3 b − 4

3.4.3 Shape of Lomax distribution The figure 3.2 below shows the shape of the Lomax distribution with parameter b=10. The graph is equivalent to the graph of beta distribution of the second kind with parameters a=1 and b=10.

Lomax distribution 10 9 8 7 6 5 4 3 2 1

0.7

0.07 0.21 0.35 0.49 0.63 0.77 0.14 0.28 0.42 0.56 0.84 0.91 0.98

0.105 0.175 0.245 0.315 0.385 0.455 0.525 0.595 0.665 0.735 0.805 0.875 0.945 0.035 Fig.3. 2: Lomax distribution

3.4.4 Log-Logistic (Fisk) distribution function Special case of beta distribution of the second kind when a=b=1 in equation (3.1)

1 푓 푥 = , 푥 > 0 ______(3.9) (1 + 푥)2

Equation 3.9 is the log-logistic distribution, also known as Fisk distribution.

3.4.5 Properties of Log-logistic distribution

i. rth-order moments Putting 푎 = 1 and 푏 = 1 in equation 3.2 Γ 1 + 푟 Γ 1 − 푟 퐸 푋푟 = Γ 1 Γ 1 = Γ 1 + 푟 Γ 1 − 푟 rπ = sin rπ

ii. Mean E(X) = Γ 1 + 푟 Γ 1 − 푟 = Γ 1 + 1 Γ −1 + 1 π = sin π

iii. Mode

1 − 1 Mode = = 0 1 + 1

iv. Variance

2π π 2 Var X = − sin 2π sin π

v. Skewness

Skewness = 퐸 푋3 − 3퐸 푋2 퐸 푋 + 2{퐸(푋)}3 3π 2π π π 3 = − 3 + 2 sin 3π sin 2π sin π sin π

vi. Kurtosis kurtosis = 퐸 푋4 − 4퐸 푋 퐸(푋3) + 6{퐸(푋)}2퐸 푋2 − 3{퐸(푋)}4 4π π 3π π 2 2π π 4 = − 4 + 6 − 3 sin 4π sin π sin 3π sin π sin 2π sin π

3.4.6 Shape of Log-logistic distribution

Log-logistic distribution 1.2

0.8

0.6

0.4

0.2

0.9

0.99 0.09 0.18 0.27 0.36 0.45 0.54 0.63 0.72 0.81

0.225 0.585 0.045 0.135 0.315 0.405 0.495 0.675 0.765 0.855 0.945

Fig.3. 3: Log-logistic distribution

3.5 Limiting Distribution Gamma distribution The two parameter beta distribution in equation 3.1 can be shown to approach the gamma distribution as b→∞ with scale parameter a changing with the parameter b as a= 푏β. The resultant probability density function is given by

푏β−1 푥 푥 − 푓(x; a, β, b) = 푒 푏훽 0 ≤ 푥 ______(3.10) 푎β Γ 푏β Equation (3.10) is the gamma distribution function which is covered in details under the generalized beta distribution of the second kind in chapter VI.

4 Chapter IV: The Three Parameter Beta Distributions

4.1 Introduction This chapter highlights some of the three parameter beta distribution functions used by various authors in literature. These three parameter beta distributions are special cases of the more generalized four parameter beta distributions.

4.2 Case I: Libby and Novick Three Parameter Beta Distribution Libby and Novick (1982) used the following approach in deriving the three parameter beta distribution

푋1 Let 푌0 = 푋0 and 푌1 = 푋0+푋1

⟹ 푋0 = 푌0 and 푌1(푋0 + 푋1) = 푋1

푌1푋0 + 푌1푋1 = 푋1

푌1푋0 = (1 − 푌1)푋1

푌0푌1 ∴ 푋1 = 1 − 푌1

d푥0 d푥0 1 0 d푦0 d푦1 푌0 푌1 푌0 퐽 = = = 2 d푥1 d푥1 2 1 − 푌1 1 − 푌1 1 − 푌1 d푦0 d푦1

푔 푦0,푦1 = 푓(푥0, 푥1) 퐽

푌0 = 푓(푥0) 푓(푥1) 2 1 − 푌1 푎 푏 푢 푣 푌0 −푢푥0 푎−1 −푣푥1 푏−1 = 푒 푥0 푒 푥1 . 2 Γ(a) Γ(b) 1 − 푌1 푎 푏 푢 푣 푌0 −푢푥0−푣푥1 푎−1 푏−1 = 푒 푥0 푥1 2 Γ(a)Γ(b) 1 − 푌1 푎 푏 푢 푣 푌0 −푢푥0−푣푥1 푎−1 푏−1 푔 푦0,푦1 = 푒 푥0 푥1 2 Γ a Γ b 1 − 푌1

푎 푏 푌0푌1 푏−1 푢 푣 −푢푦0−푣 푌0푌1 푌0 1−푌1 푎−1 = 푒 푦0 2 Γ(a)Γ(b) 1 − 푌1 1 − 푌1

푎 푏 푣푌1 푎+푏−1 푏−1 푢 푣 −푦0 푢+ 푦 푦 1−푌1 0 1 ∴ 푔 푦0,푦1 = 푒 푏+1 Γ(a)Γ(b) 1 − 푌1

푎 푏 휆1푌1 푎+푏−1 푏−1 푢 푣 −푢푦0 1+ 푦 푦 1−푌1 0 1 = 푒 푏+1 Γ(a)Γ(b) 1 − 푌1

푎 푏 푏−1 ∞ 푣푌1 푢 푣 푦 −푦0 푢+ 1 1−푌1 푎+푏−1 푔 푦1 = 푏+1 푒 푦0 푑푦0 Γ(a)Γ(b) 1 − 푌1 0 푥 푥 = 푐푦 ⟹ 푦 = 0 0 푐 푑푥 푑푦 = 0 푐 ∞ 푥 푎+푏−1 푑푥 푒−푥 0 푐 푐 푎 푏 푏−1 ∞ 푢 푣 푦1 −푐푦0 푎+푏−1 푔 푦1 = 푏+1 푒 푦0 푑푦0 Γ(a)Γ(b) 1 − 푌1 0 ∞ 푎+푏−1 −푥 푥 푒 푎+푏 푑푥 0 푐 Γ(a + b) 푢푎 푣푏 푦 푏−1 1 Γ a Γ b v푦 a+b 1 − 푌 푏+1 u + 1 1 1 − 푦1 푎 푏 푏−1 1 푢 푣 푦1 a+b 푏+1 B(a, b) λ 푦 1 − 푌1 uab 1 + 1 1 1 − 푦1 푏−1 2 v b 푦1 1 1 − 푦 1 − 푌 u 1 1 , 0 < 푦 < 1 B a, b a+b 1 푦1 1 + λ1 1 − 푦1

푏−1 2 푦1 1 λ b 1 − 푦 1 − 푌 1 1 1 B a, b a+b 푦1 1 + λ1 1 − 푦1

푏−1 2 푦1 1 λ b 1 − 푦 1 − 푌 푔 푦 = 1 1 1 1 B a, b a+b 푦1 1 + λ1 1 − 푦1

λ b 푦 푏−1 = 1 1 B a, b a+b b+1 푦1 (1 − 푌1) 1 + λ1 1 − 푦1

λ b 푦 푏−1 = 1 1 B a, b a+b b+1 1 − 푦1 + λ1푦1 (1 − 푌1) 1 − 푦1

b 푏−1 a−1 λ1 푦1 1 − 푌1 = a+b ______(4.1) B a, b 1 − (1 − λ1)푦1

Equation (4.1) was first used by Libby and Novick (1982) for utility function fitting and by Chen and Novick (1984) as prior in some binomial sampling model. The equation becomes standard beta distribution when c=1.

An alternative method of deriving the three parameter beta distribution introduced by Libby and Novick (1982) is by using transformation as follows: From the Classical beta distribution in equation (2.1), a third parameter c is introduced using the transformation 푐푦 x = 1 − 푦 + 푐푦

푑푥 푐(1 − 푦 + 푐푦 − [ −1 + 푐 푐푦] = 푑푦 (1 − 푦 + 푐푦)2

푐 − 푐푦 + 푐푦2 − [−푐푦 + 푐푦2] = (1 − 푦 + 푐푦)2

푐 = (1 − 푦 + 푐푦)2

We get

b−1 푐푦 a−1 푐푦 (1 − 푦 + 푐푦) 1 − 1 − 푦 + 푐푦 푐 f y; c, a, b = . B a, b (1 − 푦 + 푐푦)2

1 푐a−1푦a−1 (1 − 푦 + 푐푦 − 푐푦)b−1 푐 = . B a, b (1 − 푦 + 푐푦)푎−1 (1 − 푦 + 푐푦)푏−1 (1 − 푦 + 푐푦)2

1 푐a−1+1푦a−1(1 − 푦)b−1 = B a, b (1 − 푦 + 푐푦)푎−1+푏−1+2

1 푐a푦a−1(1 − 푦)b−1 = B a, b (1 − (1 − 푐)푦)푎+푏

for 0 < 푦 < 1, 푐 > 0, 푎 > 0, 푏 > 0

4.3 Case II: McDonald’s Three Parameter Beta I Distribution The general three parameter beta (a,b,c) can be derived from the standard beta distribution with two parameters in equation (2.1) by using the transformation 푦 푥 = 푐 푑푥 1 = 푑푦 푐

y a−1 y b−1 1 − 1 f y; a, b, c = c c . B a, b 푐

y a−1 y b−1 c 1 − c = c B a, b

b−1 a−1 y y 1 − c = , 0 < 푦 < 푐 , 푎 > 0, 푏 > 0, 푐 > 0______(4.2) ca B a, b The three parameter beta distribution in equation 4.2 is referred to as the beta of the first kind in McDonald, 1995. The Classical beta distribution corresponds to equation 4.2 when c=1.

4.4 Gamma Distribution as a Limiting Distribution We show that the three parameter generalized beta distribution in equation 4.2 approaches the gamma distribution as b→∞ as follows: Let the scale factor b changes with b as c=β 푎 + 푏 . Substituting this in equation 4.2 gives

푥 푏−1 푥푎−1 1 − β a + b 푓(푥; 푎, b, 푐) = β a + b 푎 퐵 푎, 푏

Collecting the terms yields

푥푎−1 Γ(푎 + 푏) 푥 푏−1 = 1 − Γ(푎)β푎 Γ(푏) a + b 푎 β a + b

For large values of b, the gamma function can be approximated by Stirling‘s formula

1 푏− 2휋 Γ(푏) = 푒−푏 푏 2

1 푎+푏− 2휋 Γ(푎 + 푏) 푒−푎−푏 (푎 + 푏) 2 ⟹ 푎 = 1 Γ(푏) a + b 푏− 2휋 푒−푏 푏 2 (푎 + 푏)푎

1 푏− 2휋 푒−푎 (푎 + 푏) 2 ∴ lim 1 = 1 푏→∞ 푏− 2휋 푏 2

푏−1 푥 푥 − Similarly, lim 1 − = 푒 훽 푏→∞ β(a + b)

푎−1 푥 푥 − ∴ 푓(x; a, β) = 푒 훽 0 ≤ 푥 ______(4.3) β푎 Γ 푎

Equation 4.3 is the gamma distribution function.

4.5 Case III

4.5.1 3 Parameter Beta Distribution Derived from the Beta Distribution of the Second Kind From the Classical beta distribution in equation 2.1, let cy 푥 = 1 + y

푑푥 c = 푑푦 1 + y 2

f(푦; 푐, 푎, 푏) = 푓(푥; 푎, 푏) 퐽

푥푎−1 1 − 푥 푏−1 = . 퐽 퐵 푎, 푏

푐푦 푎−1 푐푦 푏−1 1 + 푦 1 − 1 + 푦 c = . 퐵 푎, 푏 (1 + y)2

푐푎 푦푎−1 1 + 1 − 푐 푦 푏−1 = , 0 < 푦 < 1 ______(4.4) 퐵 푎, 푏 (1 + y)a+b

The three parameter beta distribution in equation 4.4 reduces to the beta distribution of the second kind when 푐 = 1. When 푐 = 2 it reduces to beta type 3 given by

2푎 푦푎−1 1 − 푦 푏−1 푓 푦; 푎, 푏 = , 0 < 푦 < 1______(4.5) 퐵 푎, 푏 (1 + y)a+b which is defined on a finite interval and may serve as an alternative to the Classical beta distribution for many practical applications. The beta type 3 was introduced and studied by Cardeńo et.al (2004)

4.5.2 Case IV An alternative form of the probability density function of the three parameter beta distribution of the second kind can be obtained by substituting 푦 = x/c , 푐 > 0 in equation 3.1,

Then we obtain f(푥; 푐, 푎, 푏) = 푓(푦; 푎, 푏) 퐽

푥 푎−1

= 푐 푥 a+b

푐퐵 푎, 푏 1 + 푐

푥푎−1 = , 0 ≤ 푥 < ∞, 푎 > 0, 푏 > 0______(4.6) 푥 a+b 푎 푐 퐵 푎, 푏 1 + 푐 The three parameter beta distribution in equation (4.6) was used as the beta of the second kind by Chotikapanich et.al (2010). It is referred to as beta of the second kind in McDonald and Xu, (1995). Equation 4.6 becomes the beta distribution of the second kind when c=1.

4.5.3 Case V A three parameter probability density function of beta distribution of the second kind can be obtained by using the transformation 푦 = 푐푥 , 푐 > 0 in equation 3.1,

Then we obtain f(푥; 푐, 푎, 푏) = 푓(푦; 푎, 푏) 퐽

푐푥 푎−1 = a+b . 푐 퐵 푎, 푏 1 + 푐푥

푐푎 푥푎−1 = a+b , 0 ≤ 푥 < ∞, ______(4.7) 퐵 푎, 푏 1 + 푐푥

푎 > 0, 푏 > 0, 푐 > 0

Equation 4.7 becomes the beta distribution of the second kind when c=1.

4.5.4 Case VI A three parameter probability density function of the Classical beta distribution can be obtained by using the transformation 푥 = 푦푐 푑푥 = 푐푦푐−1in equation 2.1, 푑푦

푦푐푎−푐 1 − 푦푐 푏−1 푓 푦; 푎, 푏, 푐 = . 푐푦푐−1 퐵 푎, 푏

푐푦푐푎−1 1 − 푦푐 푏−1 = , 0 < 푦 < 1, 푎 > 0, 푏 > 0, 푐 > 0____(4.8) 퐵 푎, 푏 Special case when c=1 in equation 4.8 gives the Classical beta distribution given in equation 2.1. Special case when a=1 in equation 4.8 gives

푐푦푐−1 1 − 푦푐 푏−1 푓 푦; 푏, 푐 = 퐵 1, 푏 = 푐푏푦푐−1 1 − 푦푐 푏−1 0 < 푦 < 1, 푎 > 0, 푏 > 0, 푐 > 0_____(4.9) Equation 4.9 is a Kumaraswamy distribution. Special case when c=1 and b=1 in equation 4.8 gives 푓 푦; 푎, 푐 = 푎푥푎−1 ______(4.10) Equation 4.10 is a power distribution function. When a=1 in equation 4.10 we get a Uniform distribution function.

5 Chapter V: The Four Parameter Beta Distributions

5.1 Introduction

This chapter provides for the construction, properties and special cases of four parameter generalized beta distribution of the first kind and second kind introduced by McDonald (1984) as an income distribution, whereas the beta distribution of the first kind was used for the same purpose more than a decade earlier by Thurow (1970). It also provides for the construction and properties of the five parameter generalized beta distributions which includes both generalized beta distribution of the first kind and second kind and many other distributions.

5.2 Generalized Four Parameter Beta Distribution of the First Kind

The generalized beta distribution of the first kind was introduced by McDonald (1984). It can be obtained from the Classical beta distribution in equation 2.1 by using the transformation

푦 푝 푥 = in equation (2.1) 푞

푑푥 푝 = 푦푝−1 푑푦 푞푝

The limits changes as follows 푥푞푝 = 푦푝 , when 푥 = 0, 푦푝 = 0 and when 푦 = 1, 푦푝 = 푞푝 The new distribution is

푓(푦; 푝, 푞, 푎, 푏) = 푓(푥; 푎, 푏) 퐽

푦 푝푎 −푝 푦 푝 푏−1 푞 1 − 푞 푝 = . 푦푝−1 퐵(푎, 푏) 푞푝

푦 푝 푎−1 푦 푝 푏−1

푝 푞 1 − 푞 1 = . 푦푝−1 퐵 푎, 푏 푞푝

푦 푝 푏−1 푝푎−1 푝 푦 1 − 푞 = , 0 ≤ 푦푝 ≤ 푞푝, ______(5.1) 푞푎푝 퐵 푎, 푏

The CDF of 5.1 is given by

푦 푝 퐵 푎, 푏, 푞 퐹 푦 = , 0 < 푦 < 푞 퐵 푎, 푏 where all the four parameters 푎, 푏, 푐 and 푞 are positive. 푞 is a scale parameter and 푎, 푏, 푝 are shape parameters.

The probability distribution function in equation 5.1 is a four parameter generalized beta distribution of the first kind. If p=q=1, the generalized beta distribution of the first kind in equation 5.1 becomes the two parameter Classical beta distribution in equation 2.1 over the interval (0,1). If p=1, in equation 4.1, we obtain a three parameter beta distribution in equation 4.2. When p=-1 and b=1 in equation 5.1, one obtain the inverse beta distribution of the first kind (Pareto type 1) with density function given by

푦−푎−1 푓(푦; 푞, 푎) = 푞−푎 퐵(푎, 1)

푎푞푎 = , for 푞 < 푦 푦1+푎 The generalized beta distribution of the first kind is supported on a bounded domain and is useful in the modeling of size phenomena, particularly the distribution of income.

5.2.1 Properties of generalized beta distribution of the first kind

i. rth-order moments To obtain the rth-order moments of generalized beta distribution of the first kind, we evaluate 푞 퐸 푌푟 = 푦푟 푓 푦; 푎, 푏 푑푦 0 푝 푏−1 푟+푝푎 −1 푦 푞 푝 푦 1 − 푞 = 푝푎 푑푦 0 푞 퐵 푎, 푏 푟 푏−1 푝 +푎 −1 푦 푝 푞 푝 푦 푝 1 − 1 푞 푟+푝푎 푟 = 푟 푞 퐵 + 푎, 푏 푑푦 푝푎 푝 +푎 푞 퐵 푎, 푏 0 푝 푟 푝 푞 퐵 푝 + 푎, 푏

푟+푝푎 푟 푞 퐵 푝 + 푎, 푏 = 푞푝푎 퐵 푎, 푏

푟 푟 푞 퐵 푝 + 푎, 푏 = 퐵 푎, 푏 푟 푟 푞 Γ 푝 + 푎 Γ 푏 Γ a + b = 푟

Γ 푝 + 푎 + 푏 Γ a Γ b 푟 푟 푞 Γ a + b Γ 푎 + 푝 푟 = 푟 , 푎 + > 0______(5.2) 푝 Γ 푎 + 푏 + 푝 Γ a

ii. Mean

The mean is given by substituting r=1 in equation 5.2 as

1 푞퐵 푎 + 푝 , 푏 퐸 푌 = 퐵 푎, 푏

푞Γ a + b Γ a + 푝 = ______(5.3) 1

Γ a + b + 푝 Γ a

iii. Mode The mode of generalized beta distribution of the first kind is given by 푑 푀표푑푒 = 푓 푦 = 0 푑푦 푝 푑 푦 푝 푏−1 푦푝푎−1 1 − = 0 푞푝푎 퐵 푎, 푏 푑푦 푞

푦 푝 푏−1 푦 푝 푏−2 −푝 푝푎 − 1 푦푝푎 −2 1 − + 푦푝푎 −1(푏 − 1) 1 − 푦푝−1 = 0 푞 푞 푞푝

푝푦푝(푏 − 1) (푝푎 − 1) − 푝 = 0 푝 푦 푞 1 − 푞 푝푦푝(푏 − 1) 푝푎 − 1 = 푞푝 − 푦푝 (푝푎 − 1)(푞푝 − 푦푝) = 푝푦푝(푏 − 1) 1 푞푝 푝푎 − 1 푝 푦 = 푝푎 + 푝푏 − 1 − 푝

iv. Variance Variance of generalized beta distribution of the first kind can be obtained from the rth- order moments as follows

Variance = 퐸 푌2 − 퐸 푌 2 2 1 2 2 푞 Γ a + p Γ b Γ a + b 푞Γ a + p Γ b Γ a + b = − 2 1 Γ a + b + p Γ a + b + p

2 1 2

Γ a + p Γ b Γ a + b Γ a + p Γ b Γ a + b = 푞2 − 2 1 Γ a + b + Γ a + b + p p 2 1 2 2 2 Γ a + p Γ b Γ a + b Γ a + p Γ b Γ a + b = 푞2 − 2 2 1 Γ a + b + p Γ a + b + p

v. Skewness Skewness can be given by

푆푘푒푤푛푒푠푠 = 퐸 푌3 − 3퐸 푌 퐸(푌2) +2 {퐸 푌 }3

3 1 2 3 2 푞 Γ a + b Γ a + 푝 푞Γ a + b Γ a + 푝 푞 Γ a + b Γ a + 푝 = − 3 3 1 2

Γ a + b + 푝 Γ a Γ a + b + 푝 Γ a Γ a + b + 푝 Γ a 1 3

푞Γ a + b Γ a + 푝 + 2 1

Γ a + b + 푝 Γ a 3 1 2

Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 = 푞3 − 3 3 1 2 Γ a + b + Γ a Γ a + b + Γ a Γ a + b + Γ a 푝 푝 푝 1 3

Γ a + b Γ a + 푝 + 2 1

Γ a + b + 푝 Γ a

vi. Kurtosis Kurtosis can be given by

Kurtosis = 퐸 푌4 − 4퐸 푌 퐸 푌3 + 6{퐸 푌 }2퐸 푌2 − 3 퐸 푌 4

4 1 3 4 3 q Γ a + b Γ a + 푝 qΓ a + b Γ a + 푝 q Γ a + b Γ a + 푝 = − 4 4 1 3

Γ a + b + 푝 Γ a Γ a + b + 푝 Γ a Γ a + b + 푝 Γ a 1 2 2 2 qΓ a + b Γ q + 푝 q Γ a + b Γ a + 푝 + 6 1 2

Γ a + b + 푝 Γ a Γ q + b + 푝 Γ a 1 4

qΓ a + b Γ a + 푝 − 3 1

Γ a + b + 푝 Γ a

4 1 3

Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 = q4 − 4 4 1 3 Γ a + b + Γ a Γ a + b + Γ a Γ a + b + Γ a 푝 푝 푝 1 2 2

Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 + 6 1 2

Γ a + b + 푝 Γ a Γ a + b + 푝 Γ a 1 4

Γ a + b Γ a + 푝 − 3 1

Γ a + b + 푝 Γ a

5.3 Applications of Generalized Beta Distribution of the First Kind McDonald (1984) estimated both the generalized beta of the first and second kind and both beta of the first and second kind distributions for 1970, 1975 and 1980 US family incomes and found out that the performance of the generalized beta of the first kind is comparable to the generalized gamma and beta of the second kind distribution.

5.4 Special Cases of Generalized Beta Distribution of the First Kind The four parameter probability density function of the generalized beta distribution of the first kind is very flexible and includes the following distributions:

5.4.1 Pareto type 1 distribution function

Special case of the generalized beta distribution of the first kind when p=-1 and b=1 in equation 5.1 gives

푦−푎−1 푦 푓(푦; 푞, 푎) = (1 − ( )−1)1−1 푞−푎 퐵(푎, 1) 푞

푞푎 = 퐵(푎, 1)푦푎+1

푎푞푎 = , 0 < 푞 ≤ 푦______(5.4) 푦푎+1 The cdf of equation 5.4 is given by 푞 푎 퐹 푦 = 1 − , 푦 ≥ 푞 푦

The probability distribution function in equation 5.4 is the Pareto type 1 distribution also known as classical Pareto distribution. This special case also include inverse beta of the first kind (IB1), implying Pareto (y;q,a) =IB1 (y;q,a,b=1) (McDonald 1995).

5.4.2 Properties of Pareto type 1 distribution

i. rth- order moments Putting p=-1 and b=1 in equation 5.2 푞푟 Γ a + 1 Γ 푎 − 푟 퐸(푌푟 ) = Γ 푎 + 1 − 푟 Γ a 푞푟 a = , 푎 − 푟 > 0 푎 − 푟

ii. Mean 푞Γ a + 1 Γ 푎 − 1 퐸 푌 = Γ 푎 + 1 − 1 Γ a

푞푎Γ(푎 − 1) = Γ(푎)

푞푎 푎 − 2 ! = (푎 − 1)! 푞푎 푎 − 2 ! = (푎 − 1)(푎 − 2)! 푞푎 = , 푎 > 1 푎 − 1

iii. Mode Mode of Pareto type 1 distribution can be obtained by substituting b=1 and p = −1 in the mode of generalized beta distribution of the first kind 1 − 푞−1 −1푎 − 1 1 푦 = −1푎 − 1.1 − 1 − 1 −푎 − 1 + 1 − 1 = 푞 −푎 − 1

= 푞

iv. Variance Substituting 푝 = −1 and b = 1 in the expression of variance for the generalized beta distribution of the second kind we obtain 2 Var Y = m2 − m1

푞2a 푞a 2 = − 푞2 푎 − 2 푎 − 1 푞2a a2 − 2a + 1 − 푞2푎2(푎 − 2) = (a − 2) a − 1 2 푞2a = (a − 2) a − 1 2

v. Skewness Skewness of Pareto type 1 distribution can be obtained by substituting b=1 and p = −1 in the skewness of generalized beta distribution of the first kind 푆푘푒푤푛푒푠푠 Γ a + 1 Γ a − 3 Γ a + 1 Γ a − 1 Γ a + 1 Γ a − 2 = 푞3 − 3 Γ a + 1 − 3 Γ a Γ a + 1 − 1 Γ a Γ a + 1 − 2 Γ a

Γ a + 1 Γ a − 1 3 + 2 Γ a + 1 − 1 Γ a

a 3a2 a 3 푆푘푒푤푛푒푠푠 = 푞3 − + 2 (a − 3) a − 1 (a − 2) a − 1 a a − 1 3 a − 2 − 3a2 a − 1 2(a − 3) + 2a3 a − 2 (a − 3) = 푞3 a − 1 3 a − 2 (a − 3)

(a5 − 5a4 + 9a3 − 7a2 + 2a) + (−3a5 + 15a4 − 21a3 + 9a2) +(2a5 − 10a4 + 12a3) = 푞3 a − 1 3 a − 2 (a − 3)

1 + a = 2푎푞3 a − 1 3 a − 2 (a − 3) 3 1 + a a − 2 a − 1 2 2 푠푡푎푛푑푎푟푑푖푧푒푑 푠푘푒푤푛푒푠푠 = 2푎푞3 ∗ a − 1 3 a − 2 (a − 3) 푞2a

1 + a a − 2 a − 2 a − 1 3 = 2푎푞3 ∗ a − 1 3 a − 2 (a − 3) 푞3a a

2(1 + a) a − 2 = , 푎 > 3 a − 3 a

vi. Kurtosis Kurtosis of Pareto type 1 distribution can be obtained by substituting b=1 and p = −1 in the Kurtosis of generalized beta distribution of the first kind

퐾푢푟푡표푠푖푠 Γ a + 1 Γ a − 4 Γ a + 1 Γ a − 1 Γ a + 1 Γ a − 3 = q4 − 4 Γ a + 1 − 4 Γ a Γ a + 1 − 1 Γ a Γ a + 1 − 3 Γ a

Γ a + 1 Γ a − 1 2 Γ a + 1 Γ a − 2 Γ a + 1 Γ a − 1 4 + 6 − 3 Γ a + 1 − 1 Γ a Γ a + 1 − 2 Γ a Γ a + 1 − 1 Γ a

a 4a2 6a3 3a4 = q4 − + − a − 4 (a − 1)(a − 3) (a − 1)2(a − 2) (a − 1)4

a a − 1 4 a − 2 a − 3 − 4a2 a − 1 3 a − 2 a − 4 + 6a3 a − 1 2 a − 3 a − 4 − 3a4 a − 2 a − 3 a − 4 = q4 a − 1 4 a − 2 a − 3 a − 4

(a7−9a6+32a5−58a4+57a3 − 29a2 + 6a) + 7 6 5 4 3 2 −4a + 36a − 116a + 172a − 120a + 32a + 6a7 − 54a6 + 162a5 − 186a4 + 72a3 + (−3a7 + 27a6 − 78a5 + 72a4) = q4 a − 1 4 a − 2 a − 3 a − 4

3a2 + a + 2 = 3aq4 a − 1 4 a − 2 a − 3 a − 4 푠푡푎푛푑푎푟푑푖푧푒푑 퐾푢푟푡표푠푖푠 2 3a2 + a + 2 a − 2 a − 1 2 = 3aq4 ∗ a − 1 4 a − 2 a − 3 a − 4 푞2a 3a2 + a + 2 a − 2 = 3 ∗ a − 3 a − 4 a 3(3a3 − 5a2 − 4) = a a − 3 a − 4

5.4.3 Stoppa (generalized Pareto type 1) distribution function Special case when a=1 and p=-p in equation 5.1 gives the following density function

푦 −푝 b−1 f y; p, q, b = p푞푝푏푦−푝−1 1 − , y ≥ q > 0______(5.5) 푞 The CDF of equation 5.5 is given by 푦 −푝 b F y = 1 − , y ≥ q > 0 푞 Equation 5.5 is a Stoppa distribution also known as the generalized Pareto type 1 distribution.

5.4.4 Properties of Stoppa distribution

i. rth- order moments Putting a=1 and p = −p in equation 5.2 푟 qrΓ 1 + 푏 Γ 1 − 푟 푝 퐸 푌 = 푟

Γ 1 + 푏 − 푝 Γ 1 푟 r q 푏Γ 푏 Γ 1 − 푝 = 푟 Γ 1 + 푏 − 푝

ii. Mean th The mean is given by substituting r=1 in the expression for the r -order moments 1

푞bΓ 푏 Γ 1 − 푝 E(Y) = 1 Γ 1 + 푏 − 푝

iii. Mode Putting a=1 and p = −p in the mode of generalized beta distribution of the first kind 1 − 푞−푝 −푝 − 1 푝 푦 = −푝 − 푝푏 − 1 + 푝

1 1 + 푝푏 푝 = 푞 , 푏 > 1 1 + 푝

Compared to the classical Pareto distribution, the Stoppa distribution is more flexible since it has an additional shape parameter b that allows for unimodal (for 푏 > 1) and zero-modal (for 푏 ≤ 1) densities.

iv. Variance Putting a=1 and p=-p in the variance of the generalized beta distribution of the first kind 2 1 Γ 1 − Γ b Γ 1 + b Γ2 1 − Γ2 b Γ2 1 + b p p Var Y = q2 − 2 2 1 Γ 1 + b − p Γ 1 + b − p

v. Skewness Putting a=1 and p = −p in the skewness of generalized beta distribution of the first kind

Skewness 3 1 2

Γ 1 + b Γ 1 − 푝 Γ 1 + b Γ 1 − 푝 Γ 1 + b Γ 1 − 푝 = 푞3 − 3 3 1 2 Γ 1 + b − Γ 1 Γ 1 + b − Γ 1 Γ 1 + b − Γ 1 푝 푝 푝 1 3

Γ 1 + b Γ 1 − 푝 + 2 1

Γ 1 + b − 푝 Γ 1

Skewness 3 1 2

bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 = 푞3 − 3 3 3 1 1 2 2 b − Γ b − b − Γ b − b − Γ b − 푝 푝 푝 푝 푝 푝 1 3

bΓ b Γ 1 − 푝 + 2 1 1 b − 푝 Γ b − 푝

vi. Kurtosis Putting a=1 and p = −p in the Kurtosis of generalized beta distribution of the first kind

퐾푢푟푡표푠푖푠 4 1 3

Γ 1 + b Γ 1 − 푝 Γ 1 + b Γ 1 − 푝 Γ 1 + b Γ 1 − 푝 = q4 − 4 4 1 3 Γ 1 + b − Γ 1 Γ 1 + b − Γ 1 Γ 1 + b − Γ 1 푝 푝 푝 1 2 2

Γ 1 + b Γ 1 − 푝 Γ 1 + b Γ 1 − 푝 + 6 1 2

Γ 1 + b − 푝 Γ 1 Γ 1 + b − 푝 Γ 1 1 4

Γ 1 + b Γ 1 + 푝 − 3 1

Γ 1 + b + 푝 Γ 1

4 1 3

bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 = q4 − 4 4 1 3 Γ 1 + b − Γ 1 + b − Γ 1 Γ 1 + b − 푝 푝 푝 1 2 2 Γ 1 + b Γ 1 − Γ 1 + b Γ 1 − 푝 푝 + 6 1 2 Γ 1 + b − 푎 Γ 1 + b − 푝 1 4

Γ 1 + b Γ 1 + 푝 − 3 1 Γ 1 + b + 푝

5.4.5 Kumaraswamy distributions Special case when a=1 and q=1 in equation 5.1 gives the following density function

p푦푝−1(1 − 푦푝)b−1 f(y; p, b) = B(1, b)

= pb푦푝−1(1 − 푦푝 )b−1 , b > 0______(5.6)

Equation 5.6 is a Kumaraswamy distribution. On the other hand, if b=1 in equation 5.6 then we get the power distribution 푓 y; p = pzp−1 as in equation 2.12 or equation 4.10.

100

This implies that the power distribution is a special case of Kumaraswamy distribution with parameters (a,1). If p=b=1 in equation 5.6, then the Kumaraswamy distribution becomes a uniform distribution.

5.4.6 Properties of Kumaraswamy distribution

i. rth-order moments The rth-order moments are obtained by substituting a=1 and q=1 in equation 5.2 푟 퐵 푝 + 1, 푏 퐸 푌푟 = 퐵 1, 푏 푟

Γ 푝 + 1 bΓ 푏 = 푟 Γ 푝 + 1 + 푏 푟 bΓ 푝 + 1 Γ(푏) = 푟 Γ(1 + 푝 + 푏)

ii. Mean th The mean is given by substituting r=1 in the expression for the r -order moments 1 bΓ 푝 + 1 Γ(푏) E(Y) = 1 Γ(1 + 푝 + 푏)

iii. Mode The mode of the Kumaraswamy distribution occurs at

d f y = 0 dy d pbyp−1 1 − yp b−1 = 0 dy pb p − 1 yp−2 1 − yp b−1 + b − 1 1 − yp b−2. −pyp−1. yp−1 = 0 p − 1 yp−1. y−1 1 − yp b−1 − b − 1 1 − yp b−1(1 − yp)−1pyp−1ypy−1 = 0 p − 1 − b − 1 (1 − yp)−1pyp = 0

101

b − 1 pyp p − 1 = b − 1 (1 − yp)−1pyp = (1 − yp ) p − 1 (1 − yp) = b − 1 pyp p − pyp − 1 + yp = pbyp − pyp yp −p + 1 − pb + p = 1 − p 1 − 푝 yp = 1 − 푝푏 1 1 − 푝 푝 푦 = for p ≥ 1, b ≥ 1, (p, b) ≠ (1,1) 1 − 푝푏

iv. Variance Substituting a=1 and q=1in the variance of the generalized beta distribution of the first kind we get 2 1 2 2 2 Γ 1 + p Γ b Γ 1 + b Γ 1 + p Γ b Γ 1 + b Var (Y) = − 2 2 1 Γ 1 + b + p Γ 1 + b + p 2 2 1 p ! p ! = b! b − 1 ! − b! b − 1 ! 2 1 + b ! + b ! p p

v. Skewness Putting a=1 and q = 1 in the skewness of generalized beta distribution of the first kind and is given by

푆푘푒푤푛푒푠푠 3 1 2

bΓ b Γ a + 푝 bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 = − 3 3 1 2 Γ 1 + b + 푝 Γ 1 + b + 푝 Γ 1 + b + 푝 1 3

bΓ b Γ 1 + 푝 + 2 1 Γ 1 + b + 푝

102

vi. Kurtosis Putting a=1 and q = 1 in the Kurtosis of generalized beta distribution of the first kind and is given by

퐾푢푟푡표푠푖푠 4 1 3

bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 = − 4 4 1 3 Γ 1 + b + Γ 1 + b + Γ 1 + b + 푝 푝 푝 1 2 2 1 4

bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 + 6 − 3 1 2 1 Γ 1 + b + 푝 Γ 1 + b + 푝 Γ 1 + b + 푝

5.4.7 Generalized Power distribution Function Special case when p=1 and b=1 in equation (5.1) gives the following density

푦푎−1 푓 푦; 푞, 푎 = 푞푎 퐵(푎, 1)

푎푦푎−1 = , 0 < 푦 < 푞 ______(5.7) 푞푎

Equation (5.7) is a generalized power distribution function.

5.4.8 Properties of power distribution

i. rth-order moments Putting 푝 = 1 and 푏 = 1 in equation 5.2 푞푟 Γ a + 1 Γ 푎 + 푟 퐸 푌푟 = Γ 푎 + 1 + 푟 Γ a 푞푟 aΓ a Γ 푎 + 푟 = a + r Γ 푎 + 푟 Γ a 푞푟 a = , 푎 + 푟 > 0 a + r

103

ii. Mean 푞푎 퐸 푌 = a + 1

iii. Variance Putting 푝 = 1 and b = 1 in the expression of variance for the generalized beta distribution of the second kind we obtain Γ a + 2 Γ 1 Γ a + 1 Γ2 a + 1 Γ2 1 Γ2 a + 1 Var Y = 푞2 − Γ a + 1 + 2 Γ2 a + 1 + 2 aΓ a aΓ a aΓ a = 푞2 − a + 2 (a + 1)2 a + 2 2 (a + 1)2 a + 2 aΓ a − a2Γ2 a = 푞2 (a + 1)2 a + 2 2

iv. Skewness Putting p=1 and b = 1 in the skewness of generalized beta distribution of the first kind and is given by Skewness Γ a + 1 Γ a + 3 Γ a + 1 Γ a + 1 Γ a + 1 Γ a + 2 = 푞3 − 3 Γ a + 1 + 3 Γ a Γ a + 1 + 1 Γ a Γ a + 1 + 2 Γ a

Γ a + 1 Γ a + 1 3 + 2 Γ a + 1 + 1 Γ a

a 3a2 2a3 = 푞3 − + a + 3 (a + 1) a + 2 (a + 1)3 a a + 1 3 a + 2 − 3a2(a + 1)2 a + 3 + 2a3(a + 2) a + 3 = 푞3 (a + 1)3(a + 2) a + 3 (a5+5a4+9a3+7a2 + 2a) + −3a5 − 15a4 − 21a3 − 9a2 + 2a5 + 10a4 + 12a3 = 푞3 (a + 1)3(a + 2) a + 3 2푎푞3(1 − a) = (a + 1)3(a + 2) a + 3 Standardized Skewness 3 2푎푞3(1 − a) (a + 1)2 a + 2 2 2 = . (a + 1)3(a + 2) a + 3 푞2(a + 1)2 a + 2 aΓ a − a2Γ2 a

104

3 2푎(−a3 − 3a2 + 4) 1 2 = a + 3 (a + 1)2 a + 2 aΓ a − a2Γ2 a

v. Kurtosis Putting p=1 and b = 1 in the Kurtosis of generalized beta distribution of the first kind and is given by 퐾푢푟푡표푠푖푠 Γ a + 1 Γ a + 4 Γ a + 1 Γ a + 1 Γ a + 1 Γ a + 3 = q4 − 4 Γ a + 1 + 4 Γ a Γ a + 1 + 1 Γ a Γ a + 1 + 3 Γ a

Γ a + 1 Γ a + 1 2 Γ a + 1 Γ a + 2 Γ a + 1 Γ a + 1 4 + 6 − 3 Γ a + 1 + 1 Γ a Γ a + 1 + 2 Γ a Γ a + 1 + 1 Γ a

a 4a2 6a3 3a4 = q4 − + − a + 4 (a + 1)(a + 3) a + 1 2(a + 2) a + 1 4

푆푡푎푛푑푎푟푑푖푧푒푑 퐾푢푟푡표푠푖푠 (a + 1)2 a + 2 2 2 a 4a2 = . q4 − 푞2(a + 1)2 a + 2 aΓ a − a2Γ2 a a + 4 (a + 1)(a + 3) 6a3 3a4 + − a + 1 2(a + 2) a + 1 4

5.4.9 Generalized Gamma Distribution Function

We show that the generalized beta distribution of the first kind approaches the generalized gamma distribution as b→∞ as follows:

1 Let the scale factor q changes with b as q=β(a + b)푝 . Substituting this in equation 5.1 we get

푝푎−1 푦 푝 푏−1 푝 푦 (1 − ( 1 ) ) β(a + b)푝 푓(푦; 푝, β, 푎) = 1 (β(a + b)푝 )푝푎 퐵 푎, 푏

105

Collecting the terms yields

푝 푦푝푎−1 Γ(푎 + 푏) 푦푝 푏−1 = 1 − Γ(푎)β푝푎 Γ(푏) a + b 푎 βp(a + b)

For large values of b, the gamma function can be approximated by Stirling‘s formula

1 푏− 2휋 Γ(푏) = 푒−푏 푏 2

1 푎+푏− 2휋 Γ(푎 + 푏) 푒−푎−푏 (푎 + 푏) 2 ⟹ 푎 = 1 Γ(푏) a + b 푏− 2휋 푒−푏 푏 2 (푎 + 푏)푎

1 푏− 2휋 푒−푎 (푎 + 푏) 2 ∴ lim 1 = 1 푏→∞ 푏− 2휋 푏 2

푝 푏−1 푦 푝 푦 − 훽 Similarly, lim 1 − p = 푒 푏→∞ β (a + b)

푝푎−1 푦 푝 푝 푦 − ∴ 푓(y; p, β, a) = 푒 훽 0 ≤ 푦 ______(5.8) β푝푎 Γ 푎

Equation 5.8 is the generalized Gamma (GG) distribution function. See the relationship of generalized gamma to other distribution under the GG derived from generalized beta of the second kind where it includes lognormal, Weibull, Gamma, Exponential, Normal and Pareto distributions as special or limiting cases.

5.4.10 Unit Gamma Distribution Function

Special case of generalized beta distribution of the second kind when 푎 = 훿/푝 and getting the limit as 푝 → 0 in equation 5.1 gives

106

훿 푝 −1 푦 푝 푝 푏−1 푝 푦 (1 − (푞) ) 푓 y; q, 훿, b = lim 푝→0 훿 훿 (qp)푝 B( , b) 푝

푦 푝 푏−1 푦훿−1 푝 (1 − (푞) ) = lim q훿 푝→0 훿 B(푝 , b)

푦 푝 푏−1 푦훿−1 푝 (1 − (푞) ) 훿 = lim . Γ( + b) 훿 푝→0 훿 q 푝 Γ 푝 Γ b

푦 푝 푏−1 푦훿−1 푝 (1 − (푞) ) 훿 = lim . Γ + b q훿 Γ b 푝→0 훿 푝 Γ 푝

푦훿−1 푦 푏−1 = 훿푎 ln , 0 < 푦 < 푞 ______(5.9) q훿 Γ b 푞

Equation 5.9 is the Unit gamma distribution. The unit gamma with q=1 is mentioned in Patil et al (1984) and has been used as a mixing distribution for the parameter p in the binomial, Grassia (1977).

5.5 Generalized Four Parameter Beta Distribution of the Second Kind

The generalized beta distribution of the second kind can be derived from the beta distribution of the second kind in equation 3.1 by using the transformation

푥 푝 푦 = 푞

푑푦 푝 = 푥푝−1 푑푥 푞푝

1 The new limits are as follows (푦푞푝 )푝 = 푥

107

When 푦 = 0, 푥 = 0 and when 푦 = ∞, 푥 = ∞ The new distribution is

푓(푥; 푝, 푞, 푎, 푏) = 푓(푦; 푎, 푏) 퐽

푎−1 푥 푝 (푞) 푝 = 푥푝−1 푥 a+b 푞푝 푝 퐵 푎, 푏 1 + (푞)

푝 푥푝푎−1 = , 0 < 푥 < ∞ , 푎, 푏, 푝, 푞 > 0 ______(5.10) 푝푎 푥 푝 a+b 푞 퐵(푎, 푏)(1 + (푞) )

The probability density function in equation (5.10) is a four parameter generalized beta distribution of the second kind. It is a result of the pioneering work of McDonald (1984) and Venter (1983) which led to this generalized beta of the second kind (or transformed beta) distribution. It is also known in the statistical literature as the generalized F (see, e.g., Kalbfleisch and Prentice, 1980). If p=q=1, then the generalized beta distribution of the second kind in equation 5.10 reduces to beta distribution of the second kind in equation 3.1. If p=1 in equation 5.10, one obtain the three parameter beta distribution in equation 4.6.

5.5.1 Properties of generalized beta distribution of the second kind

i. rth-order moments To obtain the rth-order moments of generalized beta distribution of the second kind, we evaluate ∞ 퐸 푌푟 = 푦푟 푓 푦; 푎, 푏 푑푦 0 ∞ 푝 푦푟+푝푎 −1 = 푑푦 푦 푎+푏 0 푝푎 푝 푞 퐵 푎, 푏 1 + (푞)

108

푟 푝 푎+ −1 1 ∞ 푝 푦 푝

= 푦 푝 푑푦 퐵 푎, 푏 0 푝푎 푎+푏 푞 1 + 푞

푟 푝 푎+ −1 푏−1 푝 푦 푝 ∞ 푝 푦 1 − ( ) 푟 1 푞 푝 푎+ = 푞 푝 푑푦 푝푎 푟 푟 푟 푞 퐵 푎, 푏 푝 푎+ 푎+ +푏− 0 푝 푦 푝 푝 푝 푞 1 + (푞)

푟 푝 푎+ 푞 푝 푟 푟 = 퐵 푎 + , 푏 − 푞푝푎 퐵 푎, 푏 푝 푝

푞푝푎+푟−푝푎 푟 푟 = 퐵 푎 + , 푏 − 퐵 푎, 푏 푝 푝

푟 푟 푟 푞 퐵 푎 + 푝 , 푏 − 푝 = 퐵 푎, 푏

푟 푟 푟 푞 Γ 푎 + 푝 Γ 푏 − 푝 = , −푝푎 < 푟 < 푝푏______(5.11) Γ(푎)Γ(푏)

ii. Mean

The mean of the generalized beta distribution of the second kind is found by substituting r=1 in equation 5.11

1 1 푞퐵 푎 + 푝 , 푏 − 푝 푚 ≡ 퐸 푌 = 1 퐵 푎, 푏

1 1 푞Γ 푎 + 푝 Γ 푏 − 푝 Γ 푎 + 푏 = . Γ 푎 + 푏 Γ 푎 Γ 푏

1 1 푞Γ 푎 + 푝 Γ 푏 − 푝 = , 푏, 푝 > 1 Γ 푎 Γ 푏

109

푟 푟 푟 푞 퐵 푎 + 푝 , 푏 − 푝 푚 ≡ 퐸 푌2 = 2 퐵 푎, 푏

2 2 2 푞 퐵 푎 + 푝 , 푏 − 푝 = 퐵 푎, 푏

iii. Mode The mode of the generalized beta distribution of the second kind occurs at

푑 푓 푦; 푝, 푞, 푎, 푏 = 0 푑푦

푑 푝 푦푝푎−1 푦 = 0 푑푦 푝푎 푝 a+b 푞 퐵 푎, 푏 (1 + (푞) )

푝 푑 푦푝푎 −1 푝푎 푦 = 0 푞 퐵 푎, 푏 푑푦 (1 + ( )푝 )a+b 푞 푦 푦 pyp−1 푝푎 − 1 푦푝푎 −1−1 (1 + ( )푝)a+b − a + b (1 + ( )푝)a+b−1 푦푝푎 −1 푞 푞 qp 2 푦 a+b 1 + ( )푝 푞 = 0

푦 푦 pyp−1 푝푎 − 1 푦푝푎 −1−1 (1 + ( )푝)a+b − a + b (1 + ( )푝 )a+b−1 푦푝푎−1 푞 푞 qp

= 0

푦 푝푎 − 1 푦푝푎 −1푦−1 (1 + ( )푝)a+b 푞

푦 푦 pypy−1 − a + b (1 + ( )푝)a+b (1 + ( )푝 )−1 푦푝푎 −1 = 0 푞 푞 qp

110

1 pyp 푝푎 − 1 = a + b 푦 푝 qp (1 + (푞) ) 푦 (qp + qp ( )푝)(푝푎 − 1) = apyp + bpyp 푞 qp 푝푎 − qp + 푦푝푝푎 − 푦푝 = apyp + bpyp 푦푝(푝푎 + 푝푏 − 푝푎 + 1) = qp (푝푎 − 1) 푞푝(푝푎 − 1) 푦푝 = (푝푏 + 1) 1 푝푎 − 1 푝 푦 = 푞 if 푝푎 ≥ 1 푝푏 + 1

iv. Variance The variance of the generalized beta distribution of the second kind is given by

Var 푌 = 퐸 푌2 − 퐸 푌 2

2 2 2 1 1 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 = 푞2 − 퐵 푎, 푏 퐵 푎, 푏

Coefficient of Variation

2 2 2 1 1 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 푞2 − 퐵 푎, 푏 퐵 푎, 푏

퐶푉 = 1 1 푞퐵 푎 + 푝 , 푏 − 푝 퐵 푎, 푏

2 2 1 1 2 퐵 푎 + , 푎 − 퐵2 푎, 푏 퐵 푎 + , 푏 − 푝 푝 푝 푝 퐵2 푎, 푏 = − 1 1 1 1 퐵 푎, 푏 퐵2 푎 + , 푏 − 퐵 푎, 푏 퐵2 푎 + , 푏 − 푝 푝 푝 푝

111

2 2

퐵 푎 + 푝 , 푏 − 푝 퐵 푎, 푏 = − 1 1 1 퐵2 푎 + , 푏 − 푝 푝

v. Skewness Skewness of the generalized beta distribution of the second kind can be given by

푆푘푒푤푛푒푠푠 = 퐸 푌3 − 3퐸 푌 퐸(푌2) +2 {퐸 푌 }3

3 3 3 1 1 2 2 2 푞 퐵 푎 + 푝 , 푏 − 푝 푞퐵 푎 + 푝 , 푏 − 푝 푞 퐵 푎 + 푝 , 푏 − 푝 = − 3 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏 1 1 3 푞퐵 푎 + 푝 , 푏 − 푝 + 2 퐵 푎, 푏

3 3 1 1 2 2 퐵 푎 + , 푏 − 퐵 푎 + , 푏 − 퐵 푎 + , 푏 − 푝 푝 푝 푝 푝 푝 = 푞3 − 3 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏

1 1 3 퐵 푎 + 푝 , 푏 − 푝 + 2 퐵 푎, 푏

퐸 푌3 − 3퐸 푌 퐸(푌2) + 2 {퐸 푌 }3 Standardized Skewness = 휍3

vi. Kurtosis Kurtosis of generalized beta distribution of the second kind can be given by 퐾푢푟푡표푠푖푠 = 퐸 푌4 − 4퐸(푌)퐸 푌3 + 6퐸{ 푌 }2퐸(푌2) −3 {퐸 푌 }4

112

4 4 4 1 1 3 3 3 푞 퐵 푎 + 푝 , 푏 − 푝 푞퐵 푎 + 푝 , 푏 − 푝 푞 퐵 푎 + 푝 , 푏 − 푝 = − 4 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏 2 1 1 2 2 2 푞퐵 푎 + 푝 , 푏 − 푝 푞 퐵 푎 + 푝 , 푏 − 푝 + 6 퐵 푎, 푏 퐵 푎, 푏

1 1 4 푞퐵 푎 + 푝 , 푏 − 푝 − 3 퐵 푎, 푏

4 4 1 1 3 3 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 = 푞4 − 4 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏

1 1 2 2 2 1 1 4 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 + 6 − 3 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏

퐸 푌4 − 4퐸(푌)퐸 푌3 + 6퐸{ 푌 }2퐸(푌2) − 3 {퐸 푌 }4 푆푡푎푛푑푎푟푑푖푧푒푑 퐾푢푟푡표푠푖푠 = 휍4

5.6 Special Case of the Generalized Beta Distribution of the Second Kind The generalized beta distribution of the second kind nests many distributions as special or limiting cases, below is a consideration of some of the distributions and their properties

5.6.1 Singh-Maddala (Burr 12) Distribution Special case of the generalized beta distribution of the second kind when a=1 in equation 5.10 gives the following density function 푝 푦푝−1 푓 푦; 푝, 푞, 푏 = 푦 푝 푝 1+b 푞 퐵 1, 푏 (1 + (푞) )

113

푝 푏푦푝−1 = , 푦 > 0______(5.12) 푝 푦 푝 1+b 푞 1 + (푞) The CDF of equation 2.1 is given by 푦 F y = 1 − 1 + ( )푝 −b 푞 Where all three parameters p, q, b are positive. q is a scale parameter and p, b are shape parameters; b only affects the right tail, whereas p affects both tails (Kleiber and Kotz, 2003). Singh Maddala distribution is known under a variety of names: Usually called Burr Type 12 distribution, or simply the Burr distribution. The Singh-Maddala distribution includes the Weibull and Fisk distributions as special cases.

5.6.2 Properties of Singh-Maddala distribution

i. rth-order moments The rth-order moments of the Singh-Maddala distribution can be obtained from the expression of the rth –order moments of the generalized beta distribution of the second kind by substituting a=1 in equation 5.11. Thus

푟 푟 푟 푞 퐵 1 + 푝 , 푏 − 푝 퐸 푌푟 = 퐵 1, 푏

푟 푟 푟 푞 Γ 1 + 푝 Γ 푏 − 푝 = , −푝 < 푟 < 푝푏 Γ b

ii. Mean

The mean is given by

1 1 푞Γ 1 + 푝 Γ 푏 − 푝 퐸 푌 = Γ b

iii. Mode The mode of the Singh-Maddala distribution is given by

114

1 푝 − 1 푝 Mode = 푞 if 푝 ≥ 1 푝푏 + 1 and at zero elsewhere. Thus the mode is decreasing with b, reflecting the fact that the right tail becomes lighter as b increases

iv. Variance 2 2 2 2 1 1 푞 퐵 1 + 푝 , 푏 − 푝 푞퐵 1 + 푝 , 푏 − 푝 Var Y = 휍2 = − 퐵 1, 푞 퐵 1, 푏

2 2 2 2 1 1 푞 bΓ 1 + 푝 Γ b − p 푞푏Γ(1 + 푝)Γ 푏 − 푝 = − bΓ(푏) bΓ 푏

2 2 2 1 2 1 Γ(b)Γ 1 + 푝 Γ 푏 − 푝 − Γ (1 + 푝)Γ 푏 − 푝 = 푞2 Γ2(푏)

Hence the coefficient of variation is given by 휍 퐶푉 = 휇 2 2 1 1 2 2 Γ b Γ 1 + 푝 Γ 푏 − 푝 − Γ 1 + 푝 Γ 푏 − 푝 푞2 Γ2 푏 퐶푉 = 1 1 푞Γ(1 + )Γ(푏 − ) 푝 푝 Γ(b) 푞 2 2 1 1 Γ b Γ 1 + Γ 푏 − − Γ2 1 + Γ2 푏 − Γ 푏 푝 푝 푝 푝 = 1 1 푞Γ(1 + 푝)Γ(푏 − 푝) Γ(b) 2 2 1 1 2 2 Γ b Γ 1 + 푝 Γ 푏 − 푝 − Γ 1 + 푝 Γ 푏 − 푝 = 1 1 Γ(1 + 푝)Γ(푏 − 푝)

115

2 2 1 1 2 2 Γ b Γ 1 + 푝 Γ 푏 − 푝 Γ 1 + 푝 Γ 푏 − 푝 = − 1 1 1 1 Γ2 1 + Γ2 푏 − Γ2 1 + Γ2 푏 − 푝 푝 푝 푝

2 2 Γ b Γ 1 + Γ 푏 − 푝 푝 − 1 2 1 2 1 Γ 1 + 푝 Γ 푏 − 푝

v. Skewness Putting a=1in the Skewness of generalized beta distribution of the second kind 푆푘푒푤푛푒푠푠 3 3 1 1 2 2 퐵 1 + 푝 , 푏 − 푝 퐵 1 + 푝 , 푏 − 푝 퐵 1 + 푝 , 푏 − 푝 = 푞3 − 3 퐵 1, 푏 퐵 1, 푏 퐵 1, 푏

1 1 3 퐵 1 + 푝 , 푏 − 푝 + 2 퐵 1, 푏푏

3 3 1 1 2 2 = 푞3 푏퐵 1 + , 푏 − − 3푏퐵 1 + , 푏 − 푏퐵 1 + , 푏 − 푝 푝 푝 푝 푝 푝

3 1 1 + 2 푏퐵 1 + , 푏 − 푝 푝

vi. Kurtosis Kurtosis of Singh-Maddala distribution can be obtained from the Kurtosis of generalized beta distribution of the second kind by letting a=1 퐾푢푟푡표푠푖푠 = 퐸 푌4 − 4퐸(푌)퐸 푌3 + 6퐸{ 푌 }2퐸(푌2) −3 {퐸 푌 }4

116

4 4 1 1 3 3 퐵 1 + 푝 , 푏 − 푝 퐵 1 + 푝 , 푏 − 푝 퐵 1 + 푝 , 푏 − 푝 = 푞4 − 4 퐵 1, 푏 퐵 1, 푏 퐵 1, 푏

1 1 2 2 2 1 1 4 퐵 1 + 푝 , 푏 − 푝 퐵 1 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 + 6 − 3 퐵 1, 푏 퐵 1, 푏 퐵 1, 푏

4 4 1 1 3 3 = 푞4 푏퐵 1 + , 푏 − − 4푏2퐵 1 + , 푏 − 퐵 1 + , 푏 − 푝 푝 푝 푝 푝 푝

2 1 1 2 2 + 6 푏퐵 1 + , 푏 − 푏퐵 1 + , 푏 − 푝 푝 푝 푝

4 1 1 − 3 푏퐵 푎 + , 푏 − 푝 푝

5.6.3 Shapes of Singh-Maddala distribution

6 (a,q,b)=(1,1,1) 5 (a,q,b)=(1,2,1) 4 (a,q,b)=(1,3,1) (a,q,b)=(2,1,1) 3 (a,q,b)=(3,1,1) 2 (a,q,b)=(0.5,2,1)

0 1

0.2 0.4 0.6 0.8

0.16 0.36 0.04 0.08 0.12 0.24 0.28 0.32 0.44 0.48 0.52 0.56 0.64 0.68 0.72 0.76 0.84 0.88 0.92 0.96

117

5.6.4 Generalized Lomax (Pareto type II) Distribution

Special case of the generalized beta distribution of the second kind when p=1, 푞 = 1 and λ 푎 = 1 in equation 5.10 gives 1 푦1(1)−1 푓 y; λ, b = 1 1.1 푦 1 1+b (휆) 퐵(1, 푏)(1 + (1) ) 휆 휆푏 = , 푦 > 0, ______(5.13) (1 + 휆푦)1+b The CDF is given by F y = 1 − (1 + 휆푦)−b

Equation 5.13 is the general expression of Lomax distribution which becomes equation 3.8 when 휆 = 1.

5.6.5 Properties of Generalized Lomax distribution

i. rth-order moments The rth-order moments of the generalized Lomax distribution can be obtained from the expression of the rth–order moments of the generalized beta distribution of the second kind by substituting 푝 = 1, 푞 = 1 , 푎 = 1 in equation 5.11. 휆 1 푟 Γ(1 + 푟)Γ(푏 − 푟) 퐸 푌푟 = 휆 Γ(b)

ii. Mean The mean is given by

1 Γ 1 + 1 Γ 푏 − 1 퐸 푌 = 휆 Γ b

118

1 푏 − 2 ! 1 = 휆 = , 푏 > 1 b − 1 (b − 2)! 휆 b − 1

iii. Mode The mode of generalized Lomax distribution is given by

1 1 − 1 Mode = = 0 휆 푏 + 1

iv. Variance The variance of generalized Lomax distribution can be obtained from the variance of generalized beta distribution of the second kind by substituting 푝 = 1, 푞 = 1 and 푎 = 1 휆

1 2 퐵 1 + 2, 푏 − 2 푞퐵 1 + 1, 푏 − 1 2 Var Y = 휆 − 퐵 1, 푏 퐵 1, 푏

1 2 1 2 2 b − 3 ! 푏 − 2 ! = 휆 − 휆 b − 1 b − 2 b − 3 ! b − 1 b − 2 !

1 2 1 2 1 = − 휆 b − 1 b − 2 b − 1 1 2 1 2 푏 − 1 − (푏 − 2) = 휆 b − 1 b − 2 (b − 1) b = , 푏 > 2 휆2 b − 1 2 b − 2 Hence the coefficient of variation is given by 휍 퐶푉 = 휇 b

휆2 b − 1 2 b − 2 퐶푉 = 1 휆 b − 1

119

휆2 b − 1 2b = 휆2 b − 1 2 b − 2

b = b − 2

v. Skewness Skewness of generalized Lomax distribution is given by 3 푠푘푒푤푛푒푠푠 = 퐸 푌3 − 3퐸 푌2 퐸 푌 + 2 퐸 푌 1 3 1 2 Γ(4)Γ(푏 − 3) Γ(3)Γ(푏 − 2) 1 1 = 휆 − 3 휆 + 2 Γ(b) Γ(b) 휆 b − 1 휆3 b − 1 3 1 6 6 2 = − + 휆3 b − 1 b − 2 b − 3 b − 1 2 b − 2 b − 1 3

1 6 b2 − 2b + 1 + 6 −b2 + 4b − 3 + 2 b2 − 5b + 6 = 휆3 b − 1 3 b − 2 b − 3 1 2b b + 1 = 휆3 b − 1 3 b − 2 b − 3

2b(b + 1) 1 푆푡푎푛푑푎푟푑푖푧푒푑 푠푘푒푤푛푒푠푠 = 3 3 . 3 휆 b − 1 b − 2 (b − 3) b

휆2 b − 1 2 b − 2

3 2b(b + 1) b − 2 = . 휆3 b − 1 3 휆3 b − 1 3 b − 2 (b − 3) b

2(b + 1) b − 2 = (b − 3) b

vi. Kurtosis Kurtosis of generalized Lomax distribution can be obtained from the kurtosis of 1 generalized beta distribution of the second kind by substituting 푝 = 1, 푞 = and 푎 = 1 휆

120

퐾푢푟푡표푠푖푠 1 4 퐵 1 + 4, 푏 − 4 퐵 1 + 1, 푏 − 1 퐵 1 + 3, 푏 − 3 = − 4 휆 퐵 1, 푏 퐵 1, 푏 퐵 1, 푏

퐵 1 + 1, 푏 − 1 2 퐵 1 + 2, 푏 − 2 퐵 1 + 1, 푏 − 1 4 + 6 − 3 퐵 1, 푏 퐵 1, 푏 퐵 1, 푏

4 1 2 = 푏퐵 5, 푏 − 4 − 4푏퐵 2, 푏 − 1 푏퐵 4, 푏 − 3 + 6 푏퐵 2, 푏 − 1 푏퐵 3, 푏 − 2 휆 4 − 3 푏퐵 2, 푏 − 1

1 4 24 4 6 = − 휆 b − 1 b − 2 b − 3 (b − 4) (b − 1) b − 1 b − 2 (b − 3) 1 2 2 1 4 + 6 − 3 (b − 1) b − 1 (b − 2) (b − 1)

1 4 24 24 12 = − + 휆 b − 1 b − 2 b − 3 (b − 4) (b − 1)2 b − 2 (b − 3) (b − 1)3(b − 2) 3 − (b − 1)4

5.6.6 Inverse Lomax Distribution

Special case of the generalized beta distribution of the second kind when b=1and p=1 in equation 5.10 gives 푎푦푎−1 푓 y; q, a = , 푦 > 0, 푞 > 0, 푎 > 0______(5.14) 푎 푦 1+a 푞 (1 + 푞) Equation 5.14 is the inverse Lomax distribution

121

5.6.7 Properties of Inverse Lomax distribution

i. rth-order moments

The rth-order moments of Inverse Lomax distribution can be obtained from the expression of the rth–order moments of the generalized beta distribution of the second kind by substituting 푝 = 1, 푏 = 1 in equation 5.11. 푞 푟 Γ(푎 + 푟)Γ(1 − 푟) 퐸 푌푟 = Γ(푎)Γ(1)

ii. Mean The mean of inverse Lomax distribution is given by

푞 Γ(푎 + 1)Γ(1 − 1) 퐸 푌 = Γ(푎)Γ(1)

iii. Mode The mode of inverse Lomax distribution is given by

푎 − 1 푀표푑푒 = 푞 2

iv. Variance The variance of inverse Lomax distribution can be obtained from the variance of generalized beta distribution of the second kind by substituting 푝 = 1, and 푏 = 1 Var Y = 푞2 푏퐵 3, 푏 − 2 − 푏퐵 2, 푏 − 1 2 Var(Y) = 푞2 푎퐵 푎 + 2,1 − 2 − 푎퐵 푎 + 1,1 − 1 2 = 푞2 푎퐵 푎 + 2, −1 − 푎퐵 푎 + 1,0 2

v. Skewness The skewness of inverse Lomax distribution can be obtained from the skewness of generalized beta distribution of the second kind by substituting 푝 = 1, and 푏 = 1

퐵 푎 + 3,1 − 3 퐵 푎 + 1,1 − 1 퐵 푎 + 2,1 − 2 퐵 푎 + 1,1 − 1 3 = 푞3 − 3 + 2 퐵 푎, 1 퐵 푎, 1 퐵 푎, 1 퐵 푎, 1

122

vi. Kurtosis The Kurtosis of inverse Lomax distribution can be obtained from the kurtosis of generalized beta distribution of the second kind by substituting 푝 = 1, and 푏 = 1

퐾푢푟푡표푠푖푠 퐵 푎 + 4,1 − 4 퐵 푎 + 1,1 − 1 퐵 푎 + 3,1 − 3 = 푞4 − 4 퐵 푎, 1 퐵 푎, 1 퐵 푎, 1

퐵 푎 + 1,1 − 1 2 퐵 푎 + 2,1 − 2 퐵 푎 + 1,1 − 1 4 + 6 − 3 퐵 푎, 1 퐵 푎, 1 퐵 푎, 1

5.6.8 Dagum (Burr type 3) Distribution

Special case of the generalized beta distribution of the second kind with b=1 in equation 5.10 gives the following density function 푝 푎푦푝푎−1 푓 y; p, q, a = 푦 , 푦 > 0, 푝 > 0, 푞 > 0, 푞푝푎 (1 + ( )푝)1+a 푞 푎 > 0______(5.15)

Equation 5.15 is the Dagum distribution. The Dagum distribution is also known as Burr Type 3 distribution in the statistic literature (Kleiber and Kotz, 2003)

5.6.9 Properties of Dagum distribution

i. rth-order moments The rth-order moments of the Dagum distribution can be obtained from the expression of the generalized beta distribution of the second kind by substituting b=1 in equation 5.11. Thus 푟 푟 푞푟 퐵 푎 + , 1 − 푝 푝 퐸 푌푟 = 퐵 푎, 1

푟 푟 푟 푞 Γ(푎 + 푝)Γ(1 − 푝) = , −푝푎 < 푟 < 푝 Γ(a)

123

ii. Mean

The mean is given by

1 1 푞Γ(푎 + 푝)Γ(1 − 푝) 퐸 푌 = Γ(a)

iii. Mode The mode of the Dagum distribution is given by

1 푝푎 − 1 푝 Mode = 푞 if 푝푎 ≥ 1 푝 + 1 and at zero elsewhere.

iv. Variance 2 2 2 2 1 1 푞 퐵 푎 + 푝 , 1 − 푝 푞퐵 푎 + 푝 , 1 − 푝 Var Y = 휍2 = − 퐵 푎, 1 퐵 푎, 1

2 2 2 2 1 1 푞 aΓ 푎 + 푝 Γ 1 − p 푞푎Γ(푎 + 푝)Γ 1 − 푝 = − aΓ(푎) aΓ 푎

2 2 2 1 2 1 Γ(a)Γ 푎 + 푝 Γ 1 − 푝 − Γ (푎 + 푝)Γ 1 − 푝 = 푞2 Γ2(푎)

Hence the coefficient of variation is 2 2 Γ(a)Γ 푎 + 푝 Γ 1 − 푝 퐶푉 = − 1 2 1 2 1 Γ (푎 + 푝)Γ 1 − 푝

124

5.6.10 Para-logistic distribution

Special case of the generalized beta distribution of the second kind with a=1 and b=p in equation 5.10 (Klugman, Panjer and Willmot, 1998)

푝 푦푝−1 푓 y; p, q = 푦 푝 푝 1+p 푞 퐵 1, 푝 (1 + (푞) )

푝2푦푝−1 = , 푦 > 0______(5.16) 푝 푦 푝 1+p 푞 (1 + (푞) )

5.6.11 General Fisk (Log-Logistic) Distribution Function

Special case of the generalized beta distribution of the second kind when a=b=1 and 푞 = 1 in equation 5.10 휆 푝 푧푝.1−1 푓 y; p, 휆 = 푝.1 1 푦 푝 1+1 휆 퐵(1,1)(1 + (1) ) 휆

휆 푝 (휆푦)푝−1 = , 푦 > 0 ______(5.17) (1 + (푦휆)푝)2

Equation 5.17 is the general form of the log logistic distribution. The Fisk (Log-logistic) distribution can also be obtained from Singh-Maddala distribution by letting b=1 and q= 1 휆 or from Dagum distribution by letting a=1 and 푞 = 1 휆

125

5.6.12 Properties of Fisk (log-logistic) distribution

i. rth-order moments The rth-order moments is obtained from the rth–order moments of the generalized beta distribution of the second kind by substituting a=b=1 in equation 5.11. Thus we get 푟 푟 퐸 푌푟 = 푞푟 퐵 1 + , 1 − 푝 푝 푟 푟 = 푞푟 Γ 1 + Γ 1 − , −푝 < 푟 < 푝 푝 푝

ii. Mean

The mean is given by

1 1 퐸 푌 = 푞Γ 1 + Γ 1 − 푝 푝

1 1 1 = 푞 Γ Γ 1 − 푝 푝 푝

iii. Mode The mode of the Fisk distribution is given by

1 푝 − 1 푝 Mode = 푞 if 푝 > 1 푝 + 1 and at zero for 푝 ≤ 1.

iv. Variance 2 2 1 1 2 Var Y = 휍2 = 푞2퐵 1 + , 1 − − 푞퐵 1 + , 1 − 푝 푝 푝 푝

126

2 2 1 1 2 = 푞2Γ 1 + Γ 1 − − 푞Γ(1 + )Γ 1 − 푝 p 푝 푝 2 2 1 1 = 푞2 Γ 1 + Γ 1 − − Γ2(1 + )Γ2 1 − 푝 푝 푝 푝

5.6.13 Fisher (F) Distribution Function

Special case of the generalized beta distribution of the second kind when 푎 = 푢 , 푏 = 2 푣 , 푝 = 1 and 푞 = 푣 in equation 5.10 2 푢 푢 1. −1 1 푦 2 푓 y; 푢, 푣 = 푢 푣 + 푢 1 2 2 1. 푣 2 푢 푣 푦

푢 퐵 2 , 2 1 + 푣 푢

푢 푢 푢 + 푣 푢 2 −1 Γ 푦 2 2 푣 = 푢+푣 ______(5.18) 푢 푣 푦푢 2 Γ 2 Γ 2 1 + 푣

Equation 5.18 is the Fisher (F) distribution.

5.6.14 Properties of F distribution

i. rth-order moments The rth-order moments is obtained from the rth–order moments of the generalized beta distribution of the second kind by substituting 푎 = 푢 , 푏 = 푣 , 푝 = 1 and 푞 = 푣 in equation 2 2 푢 5.11. Thus we get 푣 푟 푢 푣 퐵 + 푟, − 푟 푟 푢 2 2 퐸 푌 = 푢 푣 퐵 2 , 2 푣 푟 푢 푣 Γ + 푟 Γ − 푟 푢 = 푢 2 2 , − < 푟 < 1 푢 푣 2 Γ 2 Γ 2

127

ii. Mean

The mean of F distribution is given by

푣 푢 푣 푢 Γ 2 + 1 Γ 2 − 1 퐸 푌 = 푢 푣 Γ 2 Γ 2 푣 푢 푢 푣 푢 2 Γ 2 2 − 2 ! = 푢 푣 푣 Γ 2 2 − 1 2 − 2 ! 푣 2 = 푣 2 − 1 푣 = , 푣 > 2 푣 − 2

iii. Mode The mode of F distribution is given by

푢 푣 − 1 Mode = 2 푢 푣 2 + 1 푣 푢 − 2 = 푢 푣 + 2 푢 − 2 푣 = , 푢 > 2 푢 푣 + 2

iv. Variance The variance for F distribution can be given by

푣 푢 푣 ( )2퐵 + 2, − 2 푣 2 Var Y = 푢 2 2 − 푢 푣 푣 − 2 퐵(2 , 2)

128

푣 푢 푣 u + v ( )2Γ + 2 Γ − 2 Γ 푣 2 = 푢 2 2 2 − u + v 푢 푣 푣 − 2 Γ 2 Γ 2 Γ 2 푣 푢 푢 푢 푣 ( )2 + 1 Γ − 3 ! 푣 2 = 푢 2 2 2 2 − 푢 푣 푣 푣 푣 − 2 Γ 2 2 − 1 2 − 2 2 − 3 ! 푣 푢 푢 ( )2 + 1 푣 2 = 푢 2 2 − 푣 푣 푣 − 2 2 − 1 2 − 2 1 푢 + 2 1 1 2 = 푣2 푢 2 2 − 푣 − 2 푣 − 4 푣 − 2 2 2 푢 + 2 1 2 = 푣2 − 푢 푣 − 2 푣 − 4 푣 − 2 푣푢 + 2푣 − 2푢 − 4 − 푢푣 + 4푢 = 푣2 푢 푣 − 2 2 푣 − 4 푢 + 푣 − 2 = 2푣2 , 푣 > 4 푢 푣 − 2 2 푣 − 4

5.6.15 Half Student’s t distribution

Special case of the generalized beta distribution of the second kind when 푎 = 1 , 푏 = 2 푟 , 푝 = 2 and 푞 = 푟 in equation 5.10 2

1 2. −1 2 푦 2 푓 y; 푟 = 1 푟 1 2 + 2. 1 푟 푧 2 2 푟 2퐵 , 1 + 2 2 푟

1 + 푟 2Γ 2 = 1+푟 , −∞ < 푦 < ∞ ______(5.19) 푟 푦2 2 푟πΓ 2 1 + 푟

Equation 5.19 is the Half student‘s t distribution.

129

5.6.16 Logistic Distribution Function

Letting 푦 = 푒푥 and 푑푦 = 푒푥 푑푥 In equation 5.10 for the generalized beta distribution of the second kind, we get the following density function 푝푎푥 −푥 푝 푒 푥 푓 x; p, 휆 = 푥 . 푒 푝푎 푒 푝 a+b 푞 퐵(푎, 푏)(1 + ( 푞 ) )

푝 푒푝푎푥 = 푒푝푎푙표푔 (푞)퐵(푎, 푏)(1 + (푒푥 푒−log ⁡(푞))푝)a+b

푝 푒푝푎 푥−푙표푔 푞 = , −∞ < 푥 < ∞ ______(5.20) 퐵(푎, 푏) 1 + 푒푝 푥−log q a+b

Equation 5.20 is the generalized logistic distribution. McDonald and Xu (1995) referred to equation 5.20 as the density of an exponential generalized distribution. The standard logistic distribution can be obtained by setting p = q = a = b = 1 as 푒푥 푓(푥) = 1 + 푒푥 2

5.6.17 Generalized Gamma Distribution Function

We show that the generalized beta distribution of the second kind approaches the generalized gamma distribution as b→∞ as follows:

1 Let q=βb푝 . Substituting this expression in equation 5.10 yields

푝 푦푝푎−1 푓(푦; 푝, β, 푎, 푏) = 1 푝 푝푎 푦 푝 푎+푏 (βb ) 퐵 푎, 푏 (1 + ( 1 ) ) βb푝 푝 푦푝푎−1 = 1 푦 푎 β푝푎 푝 푎+푏 푏 퐵 푎, 푏 (1 + 푏 (β) )

130

Grouping the terms gives 푎+푏 푝 푦푝푎 −1 Γ(푎 + 푏 1 푓(푦; 푝, β, 푎, 푏) = 푝푎 푎 Γ(푎)β Γ 푏 b 1 푦 푝 (1 + 푏 (β) )

For large values of b, the gamma function can be approximated by Stirling‘s formula thus 1 푎+푏− 2휋 Γ(푎 + 푏) 푒−푎−푏 (푎 + 푏) 2 푎 = 1 Γ(푏)b 푏− 2휋 푒−푏 푏 2 푏푎

1 푏− 2휋 Γ(푎 + 푏) 푒−푎 (푎 + 푏) 2 lim 푎 = lim 1 = 1 b→∞ Γ(푏)b b→∞ 푏− 2휋 푏 2

푦 푝 1 − Similarly, lim = 푒 훽 b→∞ 1 푦 푝 푎+푏 (1 + 푏 (β) )

taking the limit of 푓 푦; 푝, β, 푎, 푏 as 푏 → ∞ yields

푝푎−1 푦 푝 푝 푦 − 푓 y; p, β, a = 푒 훽 , 0 ≤ 푦 ______(5.21) β푝푎 Γ 푎

Equation 5.21 is the generalized gamma (GG) distribution function equivalent to the one derived in equation 5.8.

5.6.18 Properties of Generalized Gamma distribution

i. rth-order moments To obtain the rth-order moments of the generalized gamma distribution, we evaluate

131

∞ 퐸 푌푟 = 푦푟 푓(푦; 푝, β, a)푑푦 0 ∞ 푟 푝푎−1 푦 푝 푦 푝 푧 − 훽 = 푝푎 푒 푑푦 0 β Γ 푎 ∞ (푝푎+푟)−1 푦 푝 푝 푦 − 훽 = 푝푎 푒 푑푦 0 β Γ 푎 푟 푝(푎+ )−1 푝 푟 ∞ 푝 푦 . Γ 푎 + 푧 푎 − 푝 훽 = 푟 푒 푑푦 푝(푎+ ) 푟 0 β 푝 β−푟 Γ Γ . . 푎 푎 + 푝 푟 푟 푟 푝(푎+ )−1 푝 β Γ 푎 + ∞ 푝 푦 푝 푦 − 푝 훽 = 푟 푒 푑푦 Γ 푎 푝(푎+ ) 푟 0 β 푝 Γ 푎 + 푝 The expression under the integral sign integrates to 1

푟 β푟 Γ 푎 + 푝 ∴ 퐸 푌푟 = , −pa < 푟 < ∞______(5.22) Γ 푎

ii. Mean The mean is given by

1 βΓ 푎 + 푝 퐸 푌 = Γ 푎

iii. Mode The mode of the generalized gamma distribution occurs at

d 푓(푦) = 0 dy

푝푎 −1 푦 푝 d 푝 푦 − 푒 훽 = 0 dy β푝푎 Γ 푎

푦 푝 푝−1 푦 푝 − 푝푦 − 푝푎 − 1 푦푝푎−1. 푦−1푒 훽 − . 푦푝푎 −1푒 훽 = 0 훽푝 푝푦푝 푝푎 − 1 = 훽푝

132

1 1 푝 푦 = 훽 푎 − 푝 and at zero for 푝 ≤ 1.

iv. Variance 2 2 2 1 훽 Γ 푎 + 푝 훽Γ 푎 + 푝 Var Y = − Γ(a) Γ(a)

2 2 2 2 1 훽 Γ(a)Γ 푎 + 푝 − 훽 Γ 푎 + 푝 = Γ2(a) 훽2 2 1 = Γ a Γ 푎 + − Γ2 푎 + Γ2 a 푝 푝

v. Skewness Skewness of the generalized gamma distribution can be expressed as follows Skewness = 퐸 푌3 − 3퐸 푌2 퐸 푌 + 2 퐸 푌 3 3 2 1 1 β3Γ β3Γ Γ β3Γ3 푎 + 푝 3 푎 + 푝 푎 + 푝 2 푎 + 푝 = − + Γ 푎 Γ2 푎 Γ3 푎 3 2 1 1 β3Γ2 Γ β3 Γ Γ β3Γ3 푎 푎 + 푝 − 3 Γ a 푎 + 푝 푎 + 푝 + 2 푎 + 푝 = Γ3 푎

vi. Kurtosis Kurtosis of the generalized gamma distribution can be expressed as follows Kurtosis = 퐸 푌4 − 4퐸 푌3 퐸 푌 + 6퐸 푌2 {퐸 푌 }2−3 퐸 푌 4 4 3 1 2 1 β4Γ β3Γ βΓ β2Γ β2Γ2 푎 + 푝 4 푎 + 푝 푎 + 푝 6 푎 + 푝 푎 + 푝 = − + Γ 푎 Γ 푎 Γ(a) Γ 푎 Γ2 푎 1 β4Γ4 푎 + 푝 − 3 Γ4 푎

133

4 3 1 2 1 1 Γ Γ Γ Γ Γ2 Γ4 푎 + 푝 4 푎 + 푝 푎 + 푝 6 푎 + 푝 푎 + 푝 푎 + 푝 = β4 − + − 3 Γ 푎 Γ2 푎 Γ3 푎 Γ4 푎

5.6.19 Gamma Distribution Function

Special case of the generalized gamma distribution derived in equation 5.21 when p=1 gives

푎−1 푦 푦 − 푓 y; β, a = 푒 훽 , 0 ≤ 푦 ______(5.23) β푎 Γ 푎

Equation 5.23 is the gamma distribution.

5.6.20 Properties of Gamma distribution

i. rth-order moments The rth-order moments of the Gamma distribution can be obtained from the expression of the moments of the generalized Gamma distribution by substituting p=1 in equation 5.22. Thus 푟 β푟 Γ 푎 + 푝 퐸 푌푟 = Γ 푎 β푟 Γ 푎 + 푟 = Γ 푎

ii. Mean

The mean is given by

βΓ 푎 + 1 퐸 푌 = = βa Γ 푎

134

iii. Mode The mode of the Gamma distribution occurs at

1 1 푝 Mode = 훽 푎 − = β a − 1 , a > 1 푝

iv. Variance The variance of gamma distribution can be obtained from the variance of generalized gamma distribution by substituting p=1 as follows:

훽2 Var Y = Γ a Γ 푎 + 2 − Γ2 푎 + 1 Γ2 a

훽2 = Γ a a + 1 (a)Γ 푎 − (a)Γ 푎 (a)Γ 푎 Γ2 a

훽2 = (a)Γ2 a a + 1 − (a) Γ2 a

= a훽2

v. Skewness

푆푘푒푤푛푒푠푠 = 퐸 푌3 − 3퐸 푌2 퐸 푌 + 2 퐸 푌 3

β3Γ 푎 + 3 β2Γ 푎 + 2 = − 3 . 푎β + 2 푎β 3 Γ 푎 Γ 푎

β3 a + 2 a + 1 (a)Γ 푎 3푎β3 a + 1 (a)Γ 푎 = − + 2 푎β 3 Γ 푎 Γ 푎

= β3 a + 2 a + 1 (a) − 3푎β3 a + 1 (a) + 2 푎β 3

= a3β3 + 3a2β3 + 2푎β3 − 3a3β3 − 3a2β3 + 2a3β3

= 2푎β3

The standardized skewness is given by

135

2푎β3 푆푡푎푛푑푎푟푑푖푧푒푑 푠푘푒푤푛푒푠푠 = 3 (a훽2)2

2 = 푎

vi. Kurtosis Kurtosis is given by

퐾푢푟푡표푠푖푠 = 퐸 푌4 − 4퐸 푌3 퐸 푌 + 6퐸 푌2 퐸 푌 2 − 3 퐸 푌 4 β4Γ 푎 + 4 β3Γ 푎 + 3 β2Γ 푎 + 2 = − 4 . 푎β + 6 . (푎β)2 − 3 푎β 4 Γ 푎 Γ 푎 Γ 푎

= β4 푎 + 3 푎 + 2 푎 + 1 (푎) − 4푎β4 푎 + 2 푎 + 1 (푎) + 6푎2β4(푎 + 1)(푎)

− 3푎4β4

= 푎4β4 + 6푎3β4 + 11푎2β4 + 6푎β4 − 4푎4β4 − 12푎3β4 − 8푎2β4 + 6푎4β4 + 6푎3β4

4 − 3푎4β

4 = 3푎2β + 6푎β4

The standardized kurtosis is given by

3푎2β4 + 6푎β4 푆푡푎푛푑푎푟푑푖푧푒푑 푘푢푡표푠푖푠 = 푎2β4

6 = 3 + 푎

5.6.21 Inverse Gamma distribution Special case of the generalized gamma distribution derived in equation 5.21 when 푝 = −1 gives

−푎−1 푦 −1 푦 − 푓 y; β, a = 푒 훽 , 0 ≤ 푦 β−푎 Γ 푎

푎 훽 β − = 푒 푦 , 0 ≤ 푦______(5.24) 푦1+푎 Γ 푎

Equation 5.24 is the inverse gamma distribution

136

5.6.22 Properties of Inverse Gamma distribution

i. rth-order moments The rth-order moments of the Inverse Gamma distribution can be obtained from the expression of the moments of the generalized Gamma distribution by substituting 푝 = −1 in equation 5.22. β푟 Γ 푎 − 푟 퐸 푌푟 = Γ 푎

ii. Mean The mean is given by

βΓ 푎 − 1 퐸 푌 = Γ 푎

β 푎 − 2 ! = a − 1 a − 2 !

β = a − 1

iii. Mode The mode of the inverse Gamma distribution occurs at

Mode = 훽 푎 + 1 −1 β = , a > 1 a + 1

iv. Variance The variance of inverse gamma distribution can be obtained from the variance of generalized gamma distribution by substituting 푝 = −1 as follows:

훽2 Var 푌 = Γ a Γ 푎 − 2 − Γ2 푎 − 1 Γ2 a

훽2( 푎 − 1 ! 푎 − 3 ! − 푎 − 2 ! 푎 − 2 ! = a − 1 ! a − 1 !

137

1 1 = 훽2 − a − 1 a − 2 a − 1 a − 1

훽2 = , 푎 > 2 (푎 − 1)2 a − 2

v. Skewness Skewness of inverse gamma distribution can be as follows

푆푘푒푤푛푒푠푠 = 퐸 푌3 − 3퐸 푌2 퐸 푌 + 2 퐸 푌 3

β3Γ 푎 − 3 β2Γ 푎 − 2 β β 3 = − 3 . + 2 Γ 푎 Γ 푎 a − 1 a − 1

β3 a − 4 ! 3β3 a − 3 ! 2훽3 = − + a − 1 a − 2 a − 3 a − 4 ! a − 1 2 a − 2 a − 3 ! (푎 − 1)3

β3 3β3 2훽3 = − + a − 1 a − 2 a − 3 a − 1 2 a − 2 (푎 − 1)3

푎2 − 2푎 + 1 − 3푎2 − 12푎 + 9 + 2 푎2 − 5푎 + 6 = β3 a − 1 3 a − 2 a − 3

4 = β3 a − 1 3 a − 2 a − 3

4β3 = a − 1 3 a − 2 a − 3

The standardized skewness is given by

4β3

a − 1 3 a − 2 a − 3 푆푡푎푛푑푎푟푑푖푧푒푑 푠푘푒푤푛푒푠푠 = 훽2 3 ( )2 (푎 − 1)2 a − 2

4β3 a − 1 3 a − 2 a − 2 = a − 1 3 a − 2 a − 3 β3

138

4 (a − 2) = a − 3

vi. Kurtosis Kurtosis is given by

퐾푢푟푡표푠푖푠 = 퐸 푌4 − 4퐸 푌3 퐸 푌 + 6퐸 푌2 퐸 푌 2 − 3 퐸 푌 4 β4Γ 푎 − 4 β3Γ 푎 − 3 β β2Γ 푎 − 2 β 2 β 4 = − 4 . + 6 . − 3 Γ 푎 Γ 푎 a − 1 Γ 푎 a − 1 a − 1 β4 4β4 6β4 = − + 푎 − 1 푎 − 2 푎 − 3 푎 − 4 (푎 − 1)2 푎 − 2 푎 − 3 푎 − 1 3(푎 − 2) β 4 − 3 a − 1 푎 − 1 3 − 4 푎 − 1 2 푎 − 4 + 6 푎 − 1 푎 − 3 푎 − 4 − 3 푎 − 2 푎 − 3 푎 − 4 = β4 푎 − 1 4 푎 − 2 푎 − 3 푎 − 4

푎3 − 3푎2 + 3푎 − 1 − 4푎3 + 24푎2 − 36푎 + 16 + 6푎3 − 48푎2 + 114푎 3 2 4 −72−3푎 + 27푎 − 78푎 + 72 = β 푎 − 1 4 푎 − 2 푎 − 3 푎 − 4

3푎 + 15 = β4 푎 − 1 4 푎 − 2 푎 − 3 푎 − 4

The standardized kurtosis is given by

3푎 + 15 (푎 − 1)4 a − 2 2 푆푡푎푛푑푎푟푑푖푧푒푑 푘푢푟푡표푠푖푠 = β4 . 푎 − 1 4 푎 − 2 푎 − 3 푎 − 4 훽4 3(푎2 + 3푎 − 10) = 푝 − 3 푎 − 4 3(푎 − 2)(푎 + 5) = , 푎 > 4 푎 − 3 푎 − 4

139

5.6.23 Weibull Distribution Function Special case of the generalized gamma distribution derived in equation 5.21 when a=1

푝−1 푦 푝 푝 푦 − 푓 y; p, β = 푒 훽 , 0 ≤ 푦 β푝

푦 푝 푝 − = 푦푝−1 푒 훽 ______(5.25) β푝

The cdf of the distribution in equation 5.25 is given by

푦 푝 − 퐹 푦 = 1 − 푒 훽

Equation 5.25 is the Weibull distribution with 푝 > 0 as the shape parameter and 훽 > 0 as the scale parameter.

5.6.24 Properties of Weibull distribution

i. rth-order moments The rth-order moments of the Weibull distribution can be obtained from the expression of the moments of the generalized Gamma distribution by substituting a=1 in equation 5.22. Thus 푟 β푟 Γ 1 + 푝 퐸 푌 = Γ 1 푟 = β푟 Γ 1 + 푝

ii. Mean

The mean is given by

1 β 1 퐸 푌 = βΓ 1 + = Γ 푝 p p

iii. Mode The mode occurs at

140

푑 푓 푦 = 0 푑푦

푦 푝 푑 푝 − 푦푝−1 푒 훽 = 0 푑푧 β푝

푦 푝 푝−1 푦 푝 푝 − 푝푦 − 푝 − 1 푦푝−2 푒 훽 − 푒 훽 . 푦푝−1 = 0 β푝 훽푝

푝 (푝 − 1)푦−1 = . 푦푝−1 훽푝

훽푝 (푝 − 1) = 푝푦푝

1 푝 − 1 푝 푦 = 훽 , 푝 > 0 푝

iv. Variance Var(푌) = 퐸 푌2 − (퐸(푌))2

2 1 = β2Γ 1 + − β2Γ2 1 + 푝 푝

2 = β2Γ 1 + − μ2 푝

5.6.25 Exponential distribution Function

Special case of the generalized gamma distribution in equation 5.21 with p=a=1 gives

1−1 푦 1 푧 −( )1 푓(y; p = 1, β, a = 1) = 푒 훽 0 ≤ 푦 β1Γ(1)

푦 1 −( ) = 푒 훽 , 0 ≤ 푦______(5.26) β and the cdf is given by 푦 −( ) 퐹 푌 = 1 − 푒 훽

141

Equation 5.26 is the exponential distribution. This exponential distribution can be expressed as a special case of gamma distribution in equation 5.23 with a=1 and also a special case of Weibull distribution in equation 5.25 with p=1.

5.6.26 Properties of exponential distribution

i. rth-order moments The rth-order moments for the exponential distribution can be obtained from the expression of moments of the generalized gamma distribution in equation 5.22 by substituting p = a =1

푟 β푟 Γ 1 + 퐸 푌푟 = 1 Γ 1 = β푟 Γ 1 + 푟

= β푟 rΓ 푟

ii. Mean

The mean is given by

퐸 푌 = β1(1)

= β

iii. Mode The mode of the exponential distribution occurs at

푑 푓 푦 = 0 푑푦 푦 푑 1 − = 푒 β = 0 푑푦 β 푦 1 1 − = − 푒 β = 0 β β

142

푦 − = 0 β 푦 = 0

iv. Variance From the rth-order moments given by

E(Yr) = β푟 r the variance is given by ς2 = 2β2 − (β)2

= β2

5.6.27 Chi-Squared distribution function

Special case of the generalized gamma distribution in equation 5.21 when p=1, a = n and 2 β = 2 gives

n −1 2 푦 1 1 푦 − 푓 y; n = 푒 2 , 0 ≤ 푦 n n 2Γ 2 2

n 푦 1 −1 − = 푦2 푒 2 , 0 ≤ 푦 ______(5.27) n n 22Γ 2

Equation 5.27 is the chi-squared distribution. Chi-squared distribution is a special case of the gamma distribution in equation 5.24 when 푎 = 푛 and 훽 = 2 2

5.6.28 Properties of Chi-squared distribution

i. rth-order moments The rth-order moments for the Chi-squared distribution can be obtained from the expression of moments of the generalized gamma distribution in equation 5.22 by substituting 푝 = 1, a = n and β = 2 2 n 푟 2푟 Γ + 푟 2 1 퐸 푌 = n Γ 2

143

n 푟 Γ 2 2 + 푟 = n Γ 2

ii. Mean

The mean is given by

n Γ 2 2 + 1 퐸 푌 = n Γ 2 n n Γ 2(2) 2 = n Γ 2

= n

iii. Mode The mode of the Chi-squared distribution can be obtained from the mode of generalized gamma distribution by substituting 푝 = 1, a = n and β = 2 and is given by 2 n 푦 = 2 − 1 2 = n − 2

iv. Variance The variance of Chi-squared distribution can be obtained from the variance of generalized gamma distribution by substituting 푝 = 1, a = n and β = 2 2

4 n n n Var(푌) = Γ Γ + 2 − Γ2 + 1 2 n 2 2 2 Γ 2 n n n 2 = 4 + 1 − 2 2 2

= 2n

144

5.6.29 Rayleigh distribution function

Special case of the generalized gamma distribution in equation 5.21 when p=2, a = 1 and β = α 2 gives

2−1 푦 2 2 푦 − 푓 y; α = 푒 α 2 , 0 ≤ 푦 (α 2)2Γ 1

푦 2 푦 − = 푒 α 2 , 0 ≤ 푦______(5.28) α2

The cdf is given by

푦 2 − 퐹 y = 1 − 푒 2α2 , 0 ≤ 푦

Equation 5.28 is the Rayleigh distribution. Rayleigh distribution is a special case of Weibull distribution with p=2 and β=α 2 in equation 5.25.

5.6.30 Properties of Rayleigh distribution

i. rth-order moments The rth-order moments for the Rayleigh distribution can be obtained from the expression of moments of the generalized gamma distribution in equation 5.22 by substituting p=2, a=1 and β = α 2

푟 푟 α 2 Γ 1 + 퐸 푌푟 = 2 Γ 1 푟 푟 = α 2 Γ 1 + 2

ii. Mean

The mean is given by

145

1 퐸 푌 = α 2 Γ 1 + 2

α π = 2π = α 2 2

iii. Mode The mode of the Rayleigh distribution occurs at

푑 푓 푦 = 0 푑푦

푦 2 푑 푦 − = 푒 2α2 = 0 푑푦 α2

푦 2 푦 2 1 − 2푦 − = 푒 2α2 − 푦 푒 2α2 = 0 α2 2α2 푦2 1 − = 0 α2 푦 = α

iv. Variance From the rth-order moments given by

푟 푟 E(Yr) = α 2 Γ 1 + 2

2 2 1 the variance is given by ς2 = α 2 Γ 1 + − α 2 Γ 1 + 2 2 2

4α2 − α2휋 = 2 4 − 휋 = α2 2

5.6.31 Maxwell distribution function Special case of the generalized gamma distribution in equation 5.21 when p=2, a = 3 and 2 β = α 2 gives

146

3 2( )−1 푦 2 2 푦 2 − α 2 푓 y; α = 3 푒 , 0 ≤ 푦 2( ) 3 2 Γ (α 2) 2

2 2푦 2 2 = 푒−푦 /(α 2) 3 1 Γ α 2 2 + 1 2 푦 2 2푦 − = 푒 2∝2 1 1 3 Γ 2 2 α 2 2 2 푦 2 2푦 − = 푒 2∝2 2 α3 π 2 푦 2 푦 − = 푒 2∝2 π 3 2 α

푦 2 − 2 푦2푒 2∝2 = , 0 ≤ 푦______(5.29) π α3

Equation 5.29 is the Maxwell distribution.

5.6.32 Properties of Maxwell distribution

i. rth-order moments The rth-order moments for the Maxwell distribution can be obtained from the expression of moments of the generalized gamma distribution in equation 5.22 by substituting p=2, a=3/2 and β = α 2

푟 3 푟 α 2 Γ + 퐸 푌푟 = 2 2 1 2 π

147

ii. Mean

The mean is given by

3 1 α 2 Γ + 퐸 푌 = 2 2 1 2 π

2α 2 = π

iii. Mode The mode of the Maxwell distribution occurs at

푑 푓 푦 = 0 푑푦

푦 2 푑 − = 푦2푒 2α2 = 0 푑푦

푦 2 푦 2 − 2푦 − 2푦푒 2α2 − 푦2 푒 2α2 = 0 2α2 푦3 2푦 − = 0 α2 푦 = 2α

iv. Variance From the rth-order moments given by

푟 1 푟 E(Yr) = α 2 Γ + 2 2 the variance is given by

2 3 2 2 α 2 Γ + 2α 2 푉푎푟(푌) = 2 2 − 1 π 2 π

148

8α2 = 3α2 − π 3π − 8 = α2 π

5.6.33 Half Standard normal function Special case of the generalized gamma distribution in equation 5.21 when p=2, 푎 = 1 and 2 β = 2 1 2( )−1 푦 2 2 푦 2 − 2 푓 y = 1 푒 2( ) 1 2 Γ 2 2 푦 2 2 − = 푒 2 , 0 ≤ 푦______(5.30) 2π

5.6.34 Properties of Half Standard normal distribution

i. rth-order moments The rth-order moments of the Half-standard normal distribution can be obtained from the expression of the moments of the generalized gamma distribution by substituting p=2, 1 a = and β = 2 in equation 5.22 as follows 2

푟 1 푟 2 Γ + 퐸 푌푟 = 2 2 1 Γ 2

ii. Mean The mean is given by

1 1 1 2 Γ + 퐸 푌 = 2 2 1 Γ 2

2 = π

149

iii. Mode The mode is given by 1 1 1 2 푦 = 2 − = 0 2 2

iv. Variance Variance is given by

2 1 2 Γ 2 2 + 2 2 Var Y = − 1 Γ π 2

= 1

5.6.35 Half-normal distribution function Special case of the generalized gamma distribution in equation 5.21 when 푝 = 2, 푎 = 1 2 and β = ς 2 gives 1 2( )−1 푦 2 2 푦 2 − ς 2 푓 y; ς = 1 푒 , 0 ≤ 푦 2( ) 1 2 Γ (ς 2) 2

2 2 2 = 푒−푦 /(ς 2) 1 Γ ς 2 2 푦 2 2 − = 푒 2휍2 ς 2 π 푦 2 2 − = 푒 2휍2 휍 π

푦 2 − 2 푒 2휍2 = , 0 ≤ 푦______(5.31) π 휍

Equation 5.31 is the Half-normal distribution.

150

5.6.36 Properties of Half-normal distribution

i. rth-order moments The rth-order moments of the Half-normal distribution can be obtained from the expression of the moments of the generalized gamma distribution by substituting p=2, 1 a = and β = ς 2 in equation 5.22 as follows 2

푟 1 푟 ς 2 Γ + 퐸 푌푟 = 2 2 1 Γ 2

ii. Mean

The mean is given by

1 1 ς 2 Γ + 퐸 푌 = 2 2 π ς 2 = π

iii. Mode The mode of the Half-normal distribution is given by

1 1 1 2 푦 = ς 2 − 2 2 = 0

iv. Variance Variance is given by 2 1 2 Γ 2 ς 2 2 + 2 2ς Var Y = − 1 Γ π 2 2 = ς2 1 − π

151

5.6.37 Half Log-normal distribution function Letting

푦 = (ln x − μ) and 푑푦 = 1 in equation 5.21 we get 푑푥 x

pa −1 ln 푥 −휇 푝 푝 ln 푥 − 휇 − 1 푓 x; μ, ς = 푒 훽 . 훽pa Γ 푎 푥 1 Substituting p=2, a = and β = ς 2 gives 2

(ln 푥 −휇 )2 2 − 푓 x; μ, ς = 푒 2ς2 ς푥 2π

ln 푥 −휇 2 2 − 푓 x; μ, ς = 푒 2ς2 , 푥 > 0______(5.32) ς푥 2π

Equation 5.32 is the Half Log-normal distribution. The log-normal distribution can be

1 obtained as a limiting case of the generalized gamma distribution by setting β = (ς2p2)p and a = pμ+1 as p → 0. ς2p2

5.6.38 Properties of Half Log-normal distribution

i. rth-order moments The rth-order moments of the Half Log-normal distribution is given by

1 푟휇 + 푟 2휍2 퐸 푌푟 = e 2

ii. Mean

The mean is given by

1 휇 + 휍2 퐸 푌 = e 2

152

iii. Mode The mode of the Half Log-normal distribution is given by

2 푦 = e휇 +휍

iv. Variance

2 2 Var Y = e2휇 +2휍 − e2휇 +휍

2 2 = e2휇 +휍 e휍 − 1

1 1 휇 + 휍2 2 e 2 e휍 − 1 2 퐶푉 = 1 휇 + 휍2 e 2 1 2 = e휍 − 1 2

5.7 Four Parameter Generalized Beta Distribution Function

From the beta distribution with two shape parameters a and b in equation

푎 −1 푏−1 (2.1), 푓(푥; 푎, 푏) = 푥 (1−푥) , supported on the range 0 – 1. The location and scale 퐵 푎,푏 ) parameters of the distribution can be altered by introducing two further parameters p and q representing the minimum and maximum values of the distribution respectively using the following transformation:- 푦 − 푝 푥 = 푞 − 푝

푑푥 1 = 푑푦 푞 − 푝

1 (푦 − 푝)푎−1 푦 − 푝 푏−1 1 ∴ 푓 푦; 푝, 푞, 푎, 푏 = 1 − 퐵 푎, 푏 (푞 − 푝)푎−1 푞 − 푝 푞 − 푝

1 (푦 − 푝)푎−1 푞 − 푝 − 푦 + 푝 푏−1 1 = 퐵 푎, 푏 (푞 − 푝)푎−1 푞 − 푝 푞 − 푝

153

1 1 = (푦 − 푝)푎−1 푞 − 푦 푏−1 퐵 푎, 푏 (푞 − 푝)푎−1+푏−1+1

(푦 − 푝)푎−1 푞 − 푦 푏−1 = ______(5.33) 퐵 푎, 푏 (푞 − 푝)푎+푏−1

The probability distribution function in equation 5.33 is a four parameter beta distribution function.

5.7.1 Properties of the four parameter generalized beta distribution

i. Mean ∞ ∞ 푦(푦 − 푝)푎−1 푞 − 푦 푏−1

퐸 푌 = 푦푓 푦; 푝, 푞, 푎, 푏 푑푦 = 푎+푏−1 푑푦 0 0 퐵 푎, 푏 (푞 − 푝)

1 ∞ 푎−1 푏−1 = 푎+푏−1 푦(푦 − 푝) 푞 − 푦 푑푦 퐵 푎, 푏 (푞 − 푝) 0

푎푞 + 푏푝 = 푎 + 푏

ii. Mode 푎 − 1 푞 + (푏 − 1)푝 = 푓표푟 푎 > 1, 푏 > 1 푎 + 푏 − 2

iii. Variance 푎푏(푞 − 푝)2 = 푎 + 푏 2(푎 + 푏 + 1)

154

6 Chapter VI: The Generalized Beta Distribution Based on Transformation Technique

6.1 Introduction

This chapter provides the construction, properties and special cases of the five parameter generalized beta distributions due to McDonald‘s. It gives an extension of the McDonald‘s five parameter to a five parameter weighted generalized beta distribution. It further considers a five parameter exponential generalized beta distribution function. The five parameter generalized beta distribution nests both the generalized beta distribution of the first kind and the generalized beta distribution of the second kind. It includes many distributions as special or limiting cases.

6.2 Five Parameter Generalized Beta Distribution Function Due to McDonald

McDonald and Xu (1995) introduced the five parameter generalized beta distribution and showed that it contains the distributions previously mentioned under 4-3-2 parameters, as special or limiting cases. The probability density function of the five parameter generalized beta distribution can be obtained from the classical beta distribution with two shape parameters a and b in equation (2.1), using the following transformation

푦푝 푥 = 푞푝 + 푐푦푝

푑푥 푝푦푝−1 푞푝 + 푐푦푝 − [푝푐푦푝−1(푦푝)] = 푑푦 (푞푝 + 푐푦푝 )2

푝푞푝푦푝−1 = (푞푝 + 푐푦푝 )2

푦푝푎 −푝 푦푝 푏−1 1 푝푞푝 푦푝−1 ∴ 푓 푦; 푝, 푞, 푐, 푎, 푏 = 1 − (푞푝 + 푐푦푝)푎−1 푞푝 + 푐푦푝 퐵 푎, 푏 (푞푝 + 푐푦푝 )2

155

푦푝푎 −푝 푞푝 + 푐푦푝 − 푦푝 푏−1 1 푝푞푝 푦푝−1 = (푞푝 + 푐푦푝)푎−1 푞푝 + 푐푦푝 퐵 푎, 푏 (푞푝 + 푐푦푝)2

푝푞푝푦푝푎 −푝+푝−1(푞푝 + 푐푦푝 − 푦푝 )푏−1 = 퐵 푎, 푏 (푞푝 + 푐푦푝)푎−1+푏−1+2

푝 푝 푏−1 푝 푝푎−1 푝 푏−1 푦 푦 푝푞 푦 (푞 ) 1 + 푐 푞푝 − 푞푝 = 푦푝 푎+푏 푝 푎+푏 퐵 푎, 푏 (푞 ) 1 + 푐 푞푝

푦 푏−1 푝+푝푏 −푝 푝푎 −1 푝 푝푞 푦 1 + 푐 − 1 (푞) = 푦 푎+푏 푝푎+푝푏 푝 퐵 푎, 푏 푞 1 + 푐(푞)

푦 푝 푏−1 푝푏 푝푎−1 푝푞 푦 1 − 1 − 푐 푞 = 푦 푎+푏 푝푎+푝푏 푝 푞 퐵 푎, 푏 1 + 푐(푞)

푦 푝 푏−1 푝푎 −1 푝푦 1 − 1 − 푐 푞 = ______(6.1) 푦 푝 푎+푏 푝푎 푞 퐵 푎, 푏 1 + 푐 푞

푝 for 0 < 푦푝 < 푞 , 0 ≤ 푐 ≤ 1 and 푝 > 0, 푞 > 0, 푎 > 0, 푏 > 0 1−푐

Equation 6.1 is a five parameter generalized beta distribution. The five parameter generalized beta distribution encompasses both the four parameter generalized beta distribution in equation 5.1 and 5.10 as follows:-

i. Letting c=0 in the five parameter generalized beta distribution in equation 6.1 gives a four parameter generalized beta distribution in equation 5.1 ii. Letting c=1 in the five parameter generalized beta distribution in equation 6.1 gives the four parameter generalized beta distribution in equation 5.10.

156

However, when fitting this five parameter generalized beta distribution model to 1985 family income, McDonald and Xu (1995) found that the four parameter generalized beta of the second kind subfamily is selected (in terms of likelihood and several other criteria). Thus, it appears that the five parameter generalized beta distribution does not provide additional flexibility.

6.3 Properties of the Five Parameter Generalized Beta Distribution

6.3.1 rth–order moments To obtain the rth-order moments for the five parameter generalized beta distribution, we evaluate

∞ 퐸 푌푟 = 푦푟 푓 푦; 푝, 푞, 푐, 푎, 푏 푑푦 −∞

푟 푝푎−1 푦 푝 푏−1 ∞ 푦 푝푦 (1 − (1 − 푐)(푞) ) = 푦 푑푦 −∞ 푝푎 푝 푎+푏 푞 퐵 푎, 푏 (1 + 푐(푞) )

푏−1 replacing 1 − 1 − 푐 (푦)푝 by its binomial expansion, letting 푞

푢 = (1 − 푐)(푦)푝 and collecting like terms yield 푞

푖 ∞ −푐 1 푟 푎 + 푏 푖 푎+ +푖 푟 푟 1 − 푐 푝 푏−1 퐸 푌 = 푞 푟 푢 (1 − 푢) 푑푢 푎+ +푖 푖=0 (1 − 푐) 푝 0 푖 ∞ −푐 푎 + 푏 푖 푟 1 − 푐 푟 = 푞 푟 퐵(푎 + + 푖, 푏) 푎+ +푖 푝 푖=0 1 − 푐 푝 푖 ∞ −푐 푟 푎 + 푏 푖 Γ 푎 + + 푖 Γ b 푟 1 − 푐 푝 = 푞 푟 푎+ +푖 푟 푝 푖=0 1 − 푐 Γ 푎 + 푝 + 푖 + 푏 푟 −푐 푟 푟 푎 + 푏, 푏 + ; 푞 퐵(푎 + 푝 , 푏) 푝 1 − 푐 = 푟 푎+ 푟 1 − 푐 푝 퐵(푎, 푏) 푎 + 푏 + 푝

157

푟 푟 푟 푟 푎 + , ; 푐 푞 퐵(푎 + 푝 , 푏) 푝 푝 = 푟 ______(6.2) 퐵(푎, 푏) 푎 + 푏 + 푝

Equation 6.2 is defined for all r if 푐 < 1 or for −푎 < 푟 < 푏 if 푐 = 1 푝

6.3.2 Mean The mean is given by 1 1 1 푞퐵(푎 + , 푏) 푎 + , ; 푐 푝 푝 푝 퐸(푌) = 1 푎+ 1 1 − 푐 푝 퐵(푎, 푏) 푎 + 푏 + 푝

6.3.3 Variance 2 2 2 q2B(a + , b) a + , ; c p p p Var(Y) = 2 2F1 a+ 2 1 − c p B(a, b) a + b + p 2 1 1 1 qB(a + , b) a + , ; c p p p − 1 2F1 a+ 1 1 − c p B(a, b) a + b + p

6.4 Special Cases of the Five Parameter Generalized Beta Distribution Function

6.4.1 The four parameter generalized beta distribution of the first kind This is a special case when 푐 = 0 in equation 6.1

푦 푝 푏−1 푎푝−1 푝푦 1 − 1 − 0 푞 푓(푦; 푎, 푏, 푝, 푞) = 푦 푝 푎+푏 푎푝 푞 퐵 푎, 푏 1 + 0 푞

158

푝 푏−1 푎푝−1 푦 푝푦 1 − 푞 = 푞푎푝 퐵 푎, 푏 for 0 < 푦푝 < 푞푝 , and 푝 > 0, 푞 > 0, 푎 > 0, 푏 > 0

6.4.2 The four parameter generalized beta distribution of the second kind This is a special case when 푐 = 1 in equation 6.1

푦 푝 푏−1 푝푎 −1 푝푦 1 − 1 − 1 푞 푓 푦; 푎, 푏, 푝, 푞 = 푦 푝 푎+푏 푝푎 푞 퐵 푎, 푏 1 + 1 푞

푝푦푝푎 −1 = 푦 푝 푎+푏 푝푎 푞 퐵 푎, 푏 1 + 푞

for 0 < 푦푝 < ∞ , 푝 > 0, 푞 > 0, 푎 > 0, 푏 > 0

6.4.3 The four parameter generalized beta distribution This is a special case when a = 1 in equation 6.1

푦 푝 푏−1 푝−1 푏푝푦 1 − 1 − 푐 푞 푓 푦; 푏, 푐, 푝, 푞 = 푝 1+푏 푝 푦 푞 1 + 푐 푞 for 0 < 푦푝 < ∞ , 푝 > 0, 푞 > 0, 푎 > 0, 푏 > 0

6.4.4 The four parameter generalized Logistic distribution function This is a special case when 푐 = 1, y = 푒푥 , and 퐽 = 푒푥 in equation 6.1

푏−1 (푒푥 ) 푝 푝(푒푥 )푝푎 −1 1 − 1 − 1 . (푒푥 ) 푥 푞 푓 푒 ; 푎, 푏, 푝, 푞 = 푎+푏 (푒푥 ) 푝 푝푎 푞 퐵 푎, 푏 1 + 1 푞

푝푒푎푝(푥−푙표푔푝 ) = 푞푝푎 퐵 푎, 푏 1 + 푒푝(푥−푙표푔푞 ) 푎+푏

159

for −∞ < 푥 < ∞ , 푎 > 0, 푏 > 0, 푝 > 0, 푞 > 0

6.4.5 The three parameter Inverse Beta distribution of the first kind This is a special case when 푐 = 0, and p = −1 in equation 6.1

푦 (−1) 푏−1 푎(−1)−1 (−1)푦 1 − 1 − 0 푞 푓(푦; 푎, 푏, 푞) = 푦 (−1) 푎+푏 푞푎(−1)퐵 푎, 푏 1 + 0 푞

푏−1 푎 푞 푞 1 − 푦 = 퐵 푎, 푏 푦푎+1 for 0 < 푦 < 1 , and 푎 > 0, 푏 > 0, 푞 > 0

6.4.6 The three parameter Stoppa (generalized pareto type I) distribution function This is a special case when 푐 = 0, a = 1 and p = −p in equation 6.1

푦 −푝 푏−1 1 (−푝)−1 | − 푝|푦 1 − 1 − 0 푞 푓(푦; 푏, 푝, 푞) = 푦 −푝 1+푏 1 (−푝) 푞 퐵 1, 푏 1 + 0 푞

푦 −푝 푏−1 = 푝푞푝푏푦−푝−1 1 − 푞 for 푦 ≥ 푞 > 0 , and 푝 > 0, 푏 > 0

6.4.7 The three parameter Singh-Maddala (Burr12) distribution function This is a special case when 푐 = 1, and a = 1 in equation 6.1

푦 푝 푏−1 푝(1)−1 푝푦 1 − 1 − 1 푞 푓 푦; 푏, 푝, 푞 = 푦 푝 (1)+푏 푝(1) 푞 퐵 1, 푏 1 + 1 푞

푏푝푦푝−1 = 푦 푝 1+푏 푞푝 1 + 푞

160

for 0 < 푦푝 < ∞ , 푏 > 0, 푝 > 0, 푞 > 0

6.4.8 The three parameter Dagum (Burr3) distribution function This is a special case when 푐 = 1, and b = 1 in equation 6.1

푦 푝 (1)−1 푝푎−1 푝푦 1 − 1 − 1 푞 푓 푦; 푎, 푝, 푞 = 푦 푝 푎+(1) 푝푎 푞 퐵 푎, 1 1 + 1 푞

푎푝푦푝푎−1 = 푝 푎+1 푝푎 푦 푞 1 + 푞 for 0 < 푦푝 < ∞ , 푎 > 0, 푝 > 0, 푞 > 0

6.4.9 The two parameter Classical Beta distribution This is a special case when 푐 = 0, p = 1, and q = 1 in equation 6.1

푏−1 푦 (1) 푎(1)−1 (1)푦 1 − 1 − 0 (1) 푓(푦; 푎, 푏) = 푎+푏 푦 (1) (1)푎(1)퐵 푎, 푏 1 + 0 (1)

푦푎−1 1 − 푦 푏−1 = 퐵 푎, 푏 for 0 < 푦 < 1 , and 푎 > 0, 푏 > 0

6.4.10 The two parameter Pareto type I distribution function This is a special case when 푐 = 0, p = −1, and b = 1 in equation 6.1

푦 (−1) 1−1 푎(−1)−1 (−1)푦 1 − 1 − 0 푞 푓(푦; 푎, 푞) = 푦 (−1) 푎+1 푎(−1) 푞 퐵 푎, 1 1 + 0 푞

161

푎푞푎 = 푦푎+1 for 0 < 푦 < 1 , and 푎 > 0, 푞 > 0,

6.4.11 The two parameter Kumaraswamy distribution function This is a special case when 푐 = 0, a = 1 and q = 1 in equation 6.1

푦 푝 푏−1 (1)푝−1 푝푦 1 − 1 − 0 1 푓(푦; 푏, 푝) = 푦 푝 1+푏 1 푝 1 퐵 1, 푏 1 + 0 1

= 푝푏푦푝−1 1 − 푦 푝 푏−1 for 0 < 푦푝 < 1 , and 푝 > 0, 푏 > 0

6.4.12 The two parameter generalized Power distribution function This is a special case when 푐 = 0, b = 1, and p = 1 in equation 6.1

푦 (1) 1−1 푎(1)−1 (1)푦 1 − 1 − 0 푞 푓(푦; 푎, 푞) = 푦 (1) 푎+1 푎(1) 푞 퐵 푎, 1 1 + 0 푞

푎푦푎−1 = 푞푎 for 0 < 푦 < 1 , and 푎 > 0, 푞 > 0,

6.4.13 The two parameter Beta distribution of the second kind This is a special case when 푐 = 1, p = 1, and q = 1 in equation 6.1

푏−1 푦 (1) (1)푎−1 (1)푦 1 − 1 − 1 (1) 푓 푦; 푎, 푏 = 푎+푏 푦 (1) (1)(1)푎 퐵 푎, 푏 1 + 1 (1)

푦푎−1 = 퐵 푎, 푏 1 + 푦 푎+푏 for 0 < 푦 < ∞ , 푎 > 0, 푏 > 0

162

6.4.14 The two parameter Inverse Lomax distribution function This is a special case when 푐 = 1, b = 1, and p = 1 in equation 6.1

푦 (1) 1−1 (1)푎−1 (1)푦 1 − 1 − 1 푞 푓 푦; 푎, 푞 = 푦 (1) 푎+1 (1)푎 푞 퐵 푎, 1 1 + 1 푞

푎푦푎−1 = 푎+1 푎 푦 푞 1 + 푞 for 0 < 푦 < ∞ , 푎 > 0, 푞 > 0

6.4.15 The two parameter General Fisk (Log-Logistic) distribution function 1 This is a special case when 푐 = 1, a = 1, b = 1 and q = in equation 6.1 λ

푝 (1)−1 푦 푝(1)−1 푝푦 1 − 1 − 1 1 푓 푦; λ, 푝 = λ 푝 (1)+(1) 1 푝(1) 푦 퐵 1,1 1 + 1 λ 1 λ

λ푝 λ푦 푝−1 = 1 + λ푦 푝 ퟐ for 0 < 푦푝 < ∞ , λ > 0, 푝 > 0, 푞 > 0

6.4.16 The two parameter Fisher (F) distribution function u v v This is a special case when 푐 = 1, a = , b = , p = 1 and q = in equation 6.1 2 2 u

푢 −1 (1) 2 푢 1 −1 푦 (1)푦 2 1 − 1 − 1 v u 푓 푦; 푢, 푣 = 푢 푣 + 푢 (1) 2 2 1 v 2 푢 푣 푦

u 퐵 2 , 2 1 + 1 v u

163

푢 푢 푢 + 푣 u 2 −1 2 Γ 2 v 푦 = 푢+푣 푢 푣 푦u 2 Γ 2 Γ 2 1 + v

for 0 < 푦 < ∞ , u > 0, 푣 > 0

6.4.17 The one parameter Power distribution function This is a special case when 푐 = 0, a = 1, b = 1 and q = 1 in equation 6.1

푦 푝 1−1 (1)푝−1 푝푦 1 − 1 − 0 1 푓(푦; 푝) = 푦 푝 1+1 1 푝 1 퐵 1,1 1 + 0 1

= 푝푦푝−1

for 0 < 푦 < 1 , and 푝 > 0,

6.4.18 The one parameter Wigner semicircle distribution function 3 3 This is a special case when 푐 = 0, a = , b = , p = 1 and q = 1 in equation 6.1 2 2

3 3 (1) −1 (1)−1 푦 2 2 (1)푦 1 − 1 − 0 1 푓(푦) = 3 3 3 (1) + (1) 3 3 푦 2 2 2 1 퐵 2 , 2 1 + 0 1

8 푦 1 − 푦 = , 0 < 푦 < 1 휋 Letting

푥 + 푅 1 푦 = 푎푛푑 |퐽| = 2푅 2푅 Limits changes as 푦 = 0 ⟹ 푥 > −푅 and 푦 = 1 ⟹ 푥 = 푅

푦 + 푅 푦 + 푅 8 2푅 1 − 2푅 1 푓 푥 = . 휋 2푅 2 푅2 − 푦2 = , −푅 < 푦 < 푅 푅2휋

164

6.4.19 The one parameter Lomax (pareto type II) distribution function This is a special case when 푐 = 1, a = 1, p = 1, and q = 1 in equation 6.1

푏−1 푦 (1) (1)(1)−1 (1)푦 1 − 1 − 1 (1)

푓 푦; 푏 = (1)+푏 푦 (1) (1)(1) (1) 퐵 1, 푏 1 + 1 (1)

푏 = 1 + 푦 ퟏ+푏 for 0 < 푦 < ∞ , 푏 > 0

6.4.20 The one parameter half student’s (t) distribution function 1 r This is a special case when 푐 = 1, a = , b = , p = 2 and q = r in equation 6.1 2 2

푟 −1 1 (2) 2 2 −1 푦 (2)푦 2 1 − 1 − 1 r 푓 푦; 푟 = 1 푟 1 (2) + 2 1 푟 푦 2 2 r 2 퐵 , 1 + 1 2 2 r

1 + 푟 2Γ 2 = ퟏ+풓

푟 푦ퟐ ퟐ rπ Γ 2 1 + 푟 for 0 < 푦 < ∞ , 푟 > 0

6.4.21 Uniform distribution function This is a special case when 푐 = 0, a = 1, b = 1, p = 1 and q = 1 in equation 6.1

푦 (1) 1−1 1 (1)−1 (1)푦 1 − 1 − 0 1 푓(푦) = 푦 (1) 1+1 1 (1) 1 퐵 1,1 1 + 0 1

= 1

165

6.4.22 Arcsine distribution function 1 1 This is a special case when 푐 = 0, a = , b = , p = 1 and q = 1 in equation 6.1 2 2

1 1 (1) −1 (1)−1 푦 2 2 (1)푦 1 − 1 − 0 1 푓(푦) = 1 1 1 (1) + (1) 1 1 푦 2 2 2 1 퐵 2 , 2 1 + 0 1

1 = , 0 < 푦 < 1 휋 푦(1 − 푦)

6.4.23 Triangular shaped distribution functions i. This is a special case when 푐 = 0, a = 1, b = 2, p = 1 and q = 1 in equation 6.1

푦 (1) 2−1 1 (1)−1 (1)푦 1 − 1 − 0 1 푓(푦) = 푦 (1) 1+2 1 (1) 1 퐵 1,2 1 + 0 1

= 2 1 − 푦 , 0 < 푦 < 1

ii. This is a special case when 푐 = 0, a = 2, b = 1, p = 1 and q = 1 in equation 6.1

푦 (1) 1−1 2 (1)−1 (1)푦 1 − 1 − 0 1 푓(푦) = 푦 (1) 2+1 2 (1) 1 퐵 2,1 1 + 0 1

= 2푦, 0 < 푦 < 1

6.4.24 Parabolic shaped distribution functions This is a special case when 푐 = 0, a = 2, b = 2, p = 1 and q = 1 in equation 6.1

푦 (1) 2−1 2 (1)−1 (1)푦 1 − 1 − 0 1 푓(푦) = 푦 (1) 2+2 2 (1) 1 퐵 2,2 1 + 0 1

= 6푦 1 − 푦 , 0 < 푦 < 1

166

6.4.25 Log-Logistic (Fisk) distribution function This is a special case when 푐 = 1, a = 1, b = 1, p = 1, and q = 1 in equation 6.1

(1)−1 푦 (1) (1)(1)−1 (1)푦 1 − 1 − 1 (1)

푓 푦 = (1)+(1) 푦 (1) (1)(1) (1) 퐵 1,1 1 + 1 (1)

1 = 1 + 푦 ퟐ for 0 < 푦 < ∞

6.4.26 Logistic distribution function This is a special case when 푐 = 1, a = 1, b = 1, p = 1, q = 1, y = 푒푥 , and 퐽 = 푒푥 in equation 6.1

(1)−1 (푒푥 ) (1) 푥 (1)(1)−1 푥 (1)(푒 ) 1 − 1 − 1 (1) . (푒 )

푓 푥 = (1)+(1) (푒푥 ) (1) (1)(1)(1)퐵 1,1 1 + 1 (1)

푒푥 = , for − ∞ < 푥 < ∞ 1 + 푒푥 2

167

The Five Parameter Beta Distribution and its Special Cases Diagrammatically

5 Parameter Generalized beta

4 Parameter Generalized beta Generalized Generalized beta st nd of the 1 kind beta of the 2 kind

Singh Maddala 3 Parameter Inverse beta Stoppa Dagum (Burr3) (Burr12)

Beta of the Pareto Kumaraswamy Generalized Beta of the 1st kind type I Power 2nd kind 2 Parameter

Inverse General Fisher F Lomax Fisk (Log logistic)

Power Wigner Lomax (Pareto Half student‘s t Uniform 1 Parameter Semicircle type II)

Fisk (Log Arcsine Triangular Parabolic Logistic logistic)

168

6.5 Construction of the Five Parameter Weighted Generalized Beta Distribution Function

Extending the McDonald‘s five parameter model by introducing a sixth parameter, k gives a six parameter weighted generalized beta distribution as follows.

Let

푦 푝 푏−1 푎푝 +푘−1 푝푦 1 − 1 − 푐 푞 푓(푦; 푎, 푏, 푐, 푘, 푝, 푞) = ______(6.3) 푝 푎+푏 푎푝+푘 푘 푘 푦 푞 퐵 푎 + 푝 , 푏 − 푝 1 + 푐 푞

푝 for 0 < 푦푝 < 푞 , 0 ≤ 푐 ≤ 1 and 푝 > 0, 푞 > 0, 푎 + 푘 > 0, 푏 − 푘 > 0 1−푐 푝 푝

The Weighted generalized beta distribution of the second kind with polynomial weight function 푤 푦 = 푦푘 is a special case of equation 6.3 when 푐 =1. Alternatively, it can be obtained from the generalized beta distribution of the second kind by adding some weight to the parameters 푎 and 푏.

푘 푘 Let 푎 = 푎 + and 푏 = 푏 − in equation 4.1 of generalized beta distribution of the 푝 푞 first kind.

푘 푝 푎+ −1 푝 푦 푝 푓 푦; 푝, 푞, 푎, 푏, 푘 = 푘 푎+푏 푝 푎+ 푝 푝 푘 푘 푦 푞 퐵 푎 + 푝 , 푏 − 푝 1 + 푞

푝 푦푎푝+푘−1 = ______(6.4) 푝 푎+푏 푝푎 +푘 푘 푘 푦 푞 퐵 푎 + 푝 , 푏 − 푝 1 + 푞

푦 > 0, 푝, 푞, 푎, 푏 > 0, and − 푝푎 < 푘 < 푝푏

169

Equation 6.4 is a very flexible five parameter weighted generalized beta distribution of the second kind presented by Yuan et al. (2012).

6.6 Properties of Five Parameter Weighted Generalized Beta Distribution

6.6.1 rth-order moments To obtain the rth-order moments for the weighted generalized beta distribution, we evaluate

∞ 퐸 푌푟 = 푦푟 푓 푦; 푝, 푞, 푐, 푎, 푏 푑푦 −∞ ∞ 푦푟 푝푦푝푎+푘−1 = 푎+푏 푑푦 −∞ 푝푎 +푘 푘 푘 푦 푝 푞 퐵 푎 + 푝 , 푏 − 푝 1 + 푐(푞)

푟 푘 푟 푘 푟 푞 퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 = ______(6.5) 푘 푘 퐵 푎 + 푝 , 푏 − 푝

6.6.2 Mean

The mean is given by 푘 1 푘 1 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 퐸(푌) = ______(6.6) 푘 푘 퐵 푎 + , 푏 − 푝 푝

6.6.3 Variance 2 푘 + 2 푘 + 2 푘 + 1 푘 − 1 퐵 푎 + 푝 , 푏 − 푝 푎 + 푝 , 푏 − 푝 Var(Y) = 푞2 − 푘 푘 푘 푘 퐵 푎 + , 푏 − 푎 + , 푏 − 푝 푝 푝 푝 The coefficient of variation is given by

170

푘 + 2 푘 + 2 푘 푘 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 퐶푉 = − 1 2 푘 + 1 푘 + 1 퐵 푎 + 푝 , 푏 − 푝

6.6.4 Skewness The skewness is given by 푆푘푒푤푛푒푠푠 = 퐸 푌3 − 3퐸 푌2 퐸 푌 + 2(퐸 푌 )3

3 푘 + 3 푘 + 3 2 푘 + 2 푘 + 2 푞 퐵 푎 + 푝 , 푏 − 푝 푞 퐵 푎 + 푝 , 푏 − 푝 = − 3 . 푘 푘 푘 푘 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 푘 1 푘 1 푘 1 푘 1 2 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 + 2 푘 푘 푘 푘 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝

6.6.5 Kurtosis The kurtosis is given by 2 4 푘푢푟푡표푠푖푠 = 퐸 푌4 − 4퐸 푌3 퐸 푌 + 6퐸 푌2 퐸 푌 − 3 퐸 푌 퐾푢푟푡표푠푖푠

4 푘 + 4 푘 + 4 푞 퐵 푎 + 푝 , 푏 − 푝 = 푘 푘 퐵 푎 + 푝 , 푏 − 푝

3 푘 + 3 푘 + 3 푘 1 푘 1 푞 퐵 푎 + 푝 , 푏 − 푝 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 − 4 . 푘 푘 푘 푘 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 2 2 푘 + 2 푘 + 2 푘 1 푘 1 푞 퐵 푎 + 푝 , 푏 − 푝 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 + 6 . 푘 푘 푘 푘 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 푘 1 푘 1 4 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝 − 3 푘 푘 퐵 푎 + 푝 , 푏 − 푝푝

171

6.7 Special Cases of the Five Parameter Weighted Generalized Beta Distribution Function

The five parameter weighted generalized beta distribution includes both the generalized beta distribution of the first and second kinds as special cases, it also includes several other weighted distributions as special or limiting cases: weighted generalized gamma, weighted beta of the second kind, weighted Singh-Maddala, weighted Dagum, weighted gamma, weighted Weibull, and weighted exponential distributions among others as shown below.

6.7.1 Weighted beta of the second distribution function This is a special case when p = 1 and q = 1 in equation 6.4

푦푎+푘−1 푓 푦; 푎, 푏, 푘 = ______(6.7) 퐵 푎 + 푘, 푏 − 푘 1 + 푦 푎+푏

푦 > 0, 푎, 푏, 푘 > 0, and − 1 < 푘 < 푏

6.7.2 Weighted Singh-Maddala (Burr12) distribution function This is a special case when a = 1 in equation 6.4

푝 푦푝+푘−1 푓 푦; 푏, 푘, 푝, 푞 = ______(6.8) 푝 1+푏 푝+푘 푘 푘 푦 푞 퐵 1 + 푝 , 푏 − 푝 1 + 푞

푦 > 0, 푝, 푞, 푏 > 0, and − 푝 < 푘 < 푝푏

6.7.3 Weighted Dagum (Burr3) distribution function This is a special case when b = 1 in equation 6.4

푝 푦푎푝+푘−1 푓 푦; 푝, 푞, 푎, 푘 = ______(6.9) 푝 푎+1 푝푎 +푘 푘 푘 푦 푞 퐵 푎 + 푝 , 1 − 푝 1 + 푞

푦 > 0, 푝, 푞, 푎 > 0, and − 푝푎 < 푘 < 푝

172

6.7.4 Weighted Generalized Fisk (Log-Logistic) distribution function This is a special case when , a = 1, b = 1 and q = 1 in equation 6.4 λ

푝푦푝+푘−1 푓 푦; 푝, λ, 푘 = ______(6.10) 1 푝+푘 푘 푘 퐵 푎 + , 1 − 1 + 휆푦푝 2 λ 푝 푝

푦 > 0, 푝, λ > 0, and − 푝 < 푘 < 푝

173

The Five Parameter Weighted Beta Distribution and its Special Cases Diagrammatically

5 Parameter Weighted Generalized Beta2

4 Parameter Weighted Singh Weighted Dagum Maddala (Burr12) (Burr3)

3 Parameter Weighted Beta of the 2nd kind

Weighted 2 Parameter Generalized Fisk (log-logistic)

174

6.8 Construction of the Five Parameter Exponential Generalized Beta Distribution Function

This section considers a five parameter exponential generalized beta distribution function, its construction, properties and special cases. If a random variable 푌 follows the generalized beta distribution given in equation 6.1 with parameters 푎, 푏, 푐, 푝, 푞 , then the random variable 푍 = 퐼푛 푌 is said to be distributed as an exponential generalized beta, with the pdf given by Let 푦 = 푒푧 and 푑푦 = 푒푧 푑푧 푏−1 푒푧 푝 푧 푝푎 −1 푧 푝(푒 ) 1 − 1 − 푐 푞 . 푒 푓(푧; 푎, 푏, 푐, 푝, 푞) = 푒푧 푝 푎+푏 푝푎 푞 퐵 푎, 푏 1 + 푐 푞 푏−1 푒푧 푝 푧 푝푎 푝 푒 1 − 1 − 푐 푞 = ______(6.11) 푒푧 푝 푎+푏 푞푝푎 퐵 푎, 푏 1 + 푐 푞

1 Let 푝 = and 푞 = 푒훿 in equation 6.11 휍

1 푏−1 1 푧 1 푎 푒 휍 (푒푧 )휍 1 − 1 − 푐 휍 푒훿 푓 푧; 푎, 푏, 푐, 휍, 훿 = 1 푎+푏 1 푧 푎 푒 휍 (푒훿 )휍 퐵 푎, 푏 1 + 푐 푒훿

푎 푧−훿 푧−훿 푏−1 푒 휍 1 − 1 − 푐 푒 휍 = ______(6.12) 푧−훿 푎+푏

휍퐵 푎, 푏 1 + 푐푒 휍

175

푧 − 훿 1 for − ∞ < < 퐼푛 휍 1 − 푐

Equation 6.12 is the five parameter exponential generalized beta distribution function.

6.9 Properties of Five Parameter Exponential Generalized Beta Distribution

6.9.1 Moment-generating function The moment-generating function of the exponential generalized beta distribution variates can be obtained by using the substitution 푢 = 푒(푧−훿)/휍 in 퐸(푒푡푧 ) and combining terms:

푀 푡 = 퐸(푡푍)

= 푒훿푡 퐸(푌휍푡 ; 푝 = 1, 푞 = 1, 푐, 푎, 푏)

푒훿푡 퐵 푎 + 휍푡, 푏 푎 + 휍푡, 휍푡; 푐 = 퐹 , ___(6.13) 퐵 푎, 푏 2 1 푎 + 푏 + 휍푡

Equation 6.13 is the moment-generating function of the five parameter exponential generalized beta distribution. The other moment generating functions related to special cases can be obtained from equation 6.13 as necessary; for instance, the moment generating function for the exponential generalized beta distribution of the first kind can be obtained as a special case when 푐 = 0 and is given by

푒훿푡 퐵 푎 + 휍푡, 푏 푀(푡) = 퐵 푎, 푏

While for the exponential generalized beta distribution of the second kind can be obtained as a special case when 푐 = 1 and is given by

푒훿푡 퐵 푎 + 휍푡, 푏 − 휍푡 푀 푡 = . 퐵 푎, 푏

176

6.9.2 Mean of the exponential generalized beta distribution

The mean of the exponential generalized beta distribution can be obtained by differentiating the natural logarithm of equation 6.13, the cumulant-generating function, and substituting 푡 = 0 to yield

푐휍푎 1, 1, 푎 + 1; 푐 퐸 푍 = 훿 + 휍 휓 푎 − 휓 푏 퐹 푎 + 푏 3 1 2, 푎 + 푏 + 1;

The means for the exponential generalized beta of the first kind, exponential generalized beta of the second kind, and exponential generalized gamma, respectively, can be given by

퐸1 푍 = 훿 + 휍 휓 푎 − 휓 푎 + 푏 ,

퐸2 푍 = 훿 + 휍 휓 푎 − 휓 푏 ,

퐸3 푍 = 훿 + 휍휓 푏 ,

Where 휓() denotes the digamma function. The higher-order moments for the exponential generated beta distribution are quite involving, however, relatively simple expressions for the variance, skewness and Kurtosis for the exponential generated beta distribution of the first kind, the exponential generated beta distribution of the second kind and the exponential generated gamma distribution can be easily obtained by differentiating the desired cumulant-generating functions.

6.10 Special Cases of Exponential Generalized Beta Distribution

Since the five parameter exponential generalized beta and the five parameter generalized beta distributions are related by the logarithmic transformation, similar ‗exponential‘ distributions can be obtained by transforming each of the distributions mentioned under

177

the five parameter generalized beta distribution in section 6.2.3 which corresponds to the special cases given below:

6.10.1 Exponential generalized beta of the first kind

This is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 0

푎 푧−훿 푧−훿 푏−1 푒 휍 1−푒 휍 푓 푧; 푎, 푏, 휍, 훿 = ______(6.14) 휍퐵 푎, 푏

푧 − 훿 for − ∞ < < 0 휍 Equation 6.14 is just an alternative representation of the generalized exponential distribution.

6.10.2 Exponential generalized beta of the second kind

This is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 1

푎 푧−훿 푒 휍 푓 푧; 푎, 푏, 휍, 훿 = ______(6.15) 푧−훿 푎+푏

휍퐵 푎, 푏 1 + 푒 휍

푧 − 훿 for − ∞ < < ∞ 휍 Equation 6.15 is just an alternative representation of the generalized logistic distribution.

178

6.10.3 Exponential generalized gamma

The exponential generalized gamma distribution can be derived as a limiting case 푏 → ∞ and 훿 ∗= 휍In푞 + 훿 using the same method given in section 5.6.17. It is defined by the following probability density function:

푎 푧−훿 푧−훿 푒 휍 푒−푒 휍 푓 푧; 푎, 휍, 훿 = ______(6.16) 휍ΓΞ 푝

푧 − 훿 for − ∞ < < ∞ 휍

Equation 6.16 is an alternative representation of the generalized Gompertz distribution.

6.10.4 Generalized Gumbell distribution

This is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 1 and a = b

푎 푧−훿 푒 휍 푓 푧; 푎, 휍, 훿 = ______(6.17) 푧−훿 2푎

휍퐵 푎, 푎 1 + 푒 휍

푧 − 훿 for − ∞ < < ∞ 휍

6.10.5 Exponential Burr 12

This is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 1 and b = 1

푎 푧−훿 푎푒 휍 푓 푧; 푎, 휍, 훿 = ______(6.18) 푧−훿 푎+1

휍 1 + 푒 휍

179

푧 − 훿 for − ∞ < < ∞ 휍

6.10.6 Exponential power distribution function

The 3 parameter exponential power distribution function is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 0 and 푏 = 1

푎 푧−훿 푎푒 휍 푓 푧; 푎, 휍, 훿 = ______(6.19) 휍 푧 − 훿 for − ∞ < < 0 휍

6.10.7 Two parameter exponential distribution function

The 2 parameter exponential distribution function is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 0, 푎 = 1 and 푏 = 1

푧−훿 푒 휍 푓 푧; 휍, 훿 = ______(6.20) 휍 푧 − 훿 for − ∞ < < 0 휍

6.10.8 One parameter exponential distribution function

The 1 parameter exponential distribution function is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 0, 푎 = 1, 푏 = 1 and 훿 = 0

푧 푒휍 푓 푧; 휍 = ______(6.21) 휍 푧 for − ∞ < < 0 휍

180

6.10.9 Exponential Fisk (logistic) distribution

The exponential fisk distribution is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 1, a = 1 and b = 1

푧−훿 푒 휍 푓 푧; 휍, 훿 = ______(6.22) 푧−훿 2

휍 1 + 푒 휍

푧 − 훿 for − ∞ < < ∞ 휍 Equation 6.22 of the exponential fisk distribution is also known as the logistic distribution.

6.10.10 Standard logistic distribution

The standard logistic distribution is a special case of the exponential generalized beta distribution in equation 6.12 when 푐 = 1, a = 1, b = 1 and 훿 = 0

푧 푒휍 푓 푧; 휍 = ______(6.23) 푧 2

휍 1 + 푒휍

푧 for − ∞ < < ∞ 휍

6.10.11 Exponential weibull distribution

The exponential weibull distribution is a special case of the exponential generalized gamma distribution in equation 6.16 when a = 1

푧−훿 푧−훿 푒 휍 푒−푒 휍 푓 푧; 푏, 휍, 훿 = ______(6.24) 휍ΓΞ 푝

푧 − 훿 for − ∞ < < ∞ 휍

181

The exponential weibull distribution is the extreme value type 1 distribution or the Gompertz distribution. Equation 6.24 of the exponential weibull distribution can also be derived as a limiting case (푏 → ∞ with 훿 = 훿∗ + 휍In푞) from the exponential burr 12 distribution.

182

The Five Parameter exponential Beta Distribution and its Special Cases Diagrammatically

5 Parameter Exponential Generalized Beta

4 Parameter Exponential Exponential Generalized Beta Generalized Beta nd of the 1st kind of the 2 kind

Exponential Generalized 3 Parameter Exponential Burr 12 Power Gumbell

2 Parameter Exponential Exponential Fisk (Logistic)

1 Parameter Exponential Standard Logistic

183

7 Chapter VII: Generalized Beta Distributions Based on Generated Distributions

7.1 Introduction This chapter looks at the new family of generalized beta distributions. The new family of generalized beta distributions is based on beta generators and can be classified as follows: . Beta generated distributions . Exponentiated generated distributions . Generalized beta generated distributions

7.2 Beta Generator Approach

The beta generated distribution was first introduced by Eugene et. al (2002) through its cumulative distribution function (cdf). The cdf of a beta distribution is defined by y 푡푎−1(1 − t)푏−1 푊 푡 = 푑푡 , 푎 > 0, 푏 > 0 0 퐵 푎, 푏 Note that 0 < 푦 < 1 since 0 < 푡 < 1 Replace 푦 by a cdf say G 푥 , of any distribution, since 0 < G 푥 < 1, for −∞ < 푥 < ∞ 1 G 푥 ∴ 푊 G 푥 = 푡푎−1(1 − t)푏−1푑푡 , 푎 > 0, 푏 > 0 퐵 푎, 푏 0 푊 G 푥 = 퐴 푐푑푓 표푓 푎 푐푑푓 = 퐴 푓푢푛푐푡푖표푛 표푓 푥 = 퐹 푥 퐹 푥 = 퐹 G 푥 푑 푑 ∴ (퐹 푥 = 퐹 G 푥 푑푥 푑푥 푓 푥 = {퐹′ G 푥 } G′ 푥 = 푓 G 푥 g 푥

184

푔(푥)[퐺(푥)]푎−1 1 − 퐺 푥 푏−1 ⟹ 푓(푥) = ______(7.1) 퐵 푎, 푏

Γ a Γ b for 0 < 퐺 푥 < 1, −∞ < 푥 < ∞ where 푎 > 0, 푏 > 0, 퐵 푎, 푏 = is the beta Γ a+b function. Equation 7.1 is the beta generator or beta generated distribution. It is also referred to as the generalized beta-F distribution (Sepanski and Kong, 2007). The equation can be used to generate a new family of beta distributions usually referred to as beta generated distributions. Jones (2004) called it generalized ith order statistic because the order statistic distribution is a special case when 푎 = 푖, 푏 = 푛 − 푖 + 1 which gives the following density function

푔(푦)[퐺(푦)]푖−1 1 − 퐺 푦 푛−푖+1 −1 푓 퐺 푦; 푎, 푏 = 퐵 푖, 푛 − 푖 + 1

Γ n + 1 = 푔(푦)[퐺(푦)]푖−1 1 − 퐺 푦 푛−푖 Γ 푖 Γ 푛 − 푖 + 1

n! = 푔(푦)[퐺(푦)]푖−1 1 − 퐺 푦 푛−푖 ______(7.2) 푖 − 1 ! 푛 − 푖 !

Equation 7.2 is the ith-order statistics generated from the beta distribution.

7.3 Various Beta Generated Distributions

7.3.1 Beta-Normal Distribution - (Eugene et. al (2002))

Eugene et. al (2002) introduced the beta-normal distribution, based on a composition of the classical beta distribution and the normal distribution. Its importance is more than just generalize the normal distribution. The beta-normal distribution generalizes the normal distribution and has flexible shapes, giving it greater applicability. Since then, many authors generalized other distributions similar to the beta-normal distribution.

The beta-normal distribution is obtained as follows:

185

Let 푥 − 휇 퐺 푥 = Φ 휍 be the cumulative density function of normal distribution with parameters 휇 and 휍, and 푥 − 휇 푔 푥 = 휙 휍 be the probability density function of the normal distribution. By using equation 7.1, the density function of beta-normal distribution is given by

푥 − 휇 푎−1 푥 − 휇 푏−1 휍−1 Φ 1 − Φ 푥 − 휇 푓 푥; 푎, 푏, 휇, 휍 = 휍 휍 휙 ___(7.3) 퐵 푎, 푏 휍 Where 푎 > 0, 푏 > 0, 휍 > 0, 휇 ∈ ℝ 푎푛푑 푥 ∈ ℝ

The parameters 푎 and 푏 are the shape parameters characterizing the skewness, kurtosis and bimodality of the beta normal distribution. The parameters 휇 and 휍 have the same role as in normal distribution where, 휇 is a location parameter and 휍 is a scale parameter that stretches out or shrinks the distribution.

Special case of equation 7.3 when 휇 = 0 and 휍 = 1 gives the standard beta-normal distribution given by

[Φ 푥 ]푎−1 1 − Φ 푥 푏−1 푓 푥; 푎, 푏, = 휙 푥 ______(7.4) 퐵 푎, 푏 Where 푎 > 0, 푏 > 0, 푥 ∈ ℝ

Equation 7.3 is the beta-normal distribution introduced by Eugene et.al who also discussed some of its properties.

7.3.2 Beta-Exponential distribution – (Nadarajah and Kotz (2006))

Nadarajah and Kotz (2006) introduced the beta-exponential distribution, based on a composition of the classical beta distribution and the exponential distribution. They discussed in their paper the expression for the rth moment, properties of the hazard function, results for the distribution of the sum of beta-exponential random variables, maximum likelihood estimation and some asymptotic results.

186

The beta-exponential distribution is obtained as follows:

Let

퐺 푥 = 1 − exp(−λx) be the cdf of exponential distribution with parameter λ, and

푔 푥 = λ exp(−λx) be the pdf of exponential distribution

By using equation 7.1, the density of the beta-exponential distribution is given by

λ exp(−λx)[1 − exp(−λx) ]푎−1 1 − 1 − exp −λx 푏−1 푓 푥; 푎, 푏, λ = 퐵 푎, 푏 λ exp(−푏λx)[1 − exp(−λx) ]푎−1 = ______(7.5) 퐵 푎, 푏

푎 > 0, 푏 > 0, λ > 0, 푥 >0

Equation 7.5 is the beta-exponential distribution introduced by Nadarajah and Kotz (2006). They also discussed some of its properties. This beta-exponential distribution contains the exponentiated-exponential distribution (Gupta and Kundu (1999)) as a special case for b=1. i.e

푓 푥; 푎, λ = 푎λ exp(−λx)[1 − exp(−λx) ]푎−1

푎 > 0, λ > 0, 푥 >0

When 푎 = 1, the beta-exponential distribution coincides with the exponential distribution with parameters 푏λ i.e

푓 푥; 푏, λ = bλ exp(−푏λx)

푏 > 0, λ > 0, 푥 >0

187

7.3.3 Beta-Weibull Distribution – (Famoye et al. (2005))

Famoye et.al (2005) introduced the beta-weibull distribution, based on a composition of the classical beta distribution and the weibull distribution. Lee et.al (2007) gave some properties of the hazard function, entropies and an application to censored data. Cordeiro et.al (2005) derived additional mathematical properties of beta-weibull such as expression for moments.

The beta-weibull distribution is obtained as follows:

Let

x c − 퐺 푥 = 1 − e β be the cdf of Weibull distribution and

푥 푐 푐 − 푔 푥 = 푥푐−1 푒 훽 β푐 be the pdf of Weibull distribution

By using equation 7.1, the density function of the beta-weibull distribution is given by

푎−1 푏−1 푥 푐 x c x c 푐 − − − 푥푐−1 푒 훽 1 − e β 1 − 1 − e β β푐 푓 푥; 푎, 푏, 푐, 훽 = 퐵 푎, 푏

푎−1 푏−1 푥 푐 x c x c 푐 − − − 푥푐−1 푒 훽 1 − e β e β β푐 = 퐵 푎, 푏

푎−1 푥 푐 x c 푐 −푏 − 푥푐−1 푒 훽 1 − e β β푐 = , 푎, 푏, 푐, β > 0______(7.6) 퐵 푎, 푏

188

Equation 7.6 is the four parameter beta-Weibull distribution introduced by Famoye et.al (2005) and studied by Lee et.al (2007).

7.3.4 Beta-Hyperbolic Secant Distribution – (Fischer and Vaughan. (2007))

Fisher and Vaughan (2007) introduced the beta-hyperbolic secant distribution, based on a composition of the classical beta distribution and the hyperbolic secant distribution. They gave some of its properties like the moments, special and limiting cases and compare it with other distributions using a real data set.

The beta-hyperbolic secant distribution is obtained as follows:

Let

2 퐺 푥 = arctan 푒푥 π be the cdf of the hyperbolic secant distribution and

1 푔 푥 = πCosh(x) be the pdf of the hyperbolic secant distribution

By using equation 7.1, the density function of the beta-hyperbolic secant is given by

푏−1 2 푎−1 2 arctan 푒푥 1 − arctan 푒푥 푓 푥; 푎, 푏, π = π π ______(7.7) 퐵 푎, 푏 π cosh x

푎 > 0, 푏 > 0 and 푥 ∈ ℝ

Equation 7.7 is the beta-hyperbolic secant distribution introduced and studied by. Fischer and Vaughan (2007).

7.3.5 Beta-Gamma – (Kong et al. (2007))

Kong et.al (2007) introduced the beta-gamma distribution, based on a composition of the classical beta distribution and the gamma distribution. They derived in their paper some

189

properties of the limit of the density function and of the hazard function. They gave an expression for the moments when the shape parameter 푎 is an integer and made an application of the beta-gamma distribution.

The beta-gamma distribution is obtained as follows:

Let

Γx ρ 퐺 푥 = λ Γ ρ be the cdf of gamma distribution where

x ρ−1 −y Γx ρ = y e dy 0 is the incomplete gamma function and

ρ−1 x x − 푔 푥 = e λ λ is the pdf of the gamma function.

By using equation 7.1, the density function of the beta-gamma distribution is given by

푏−1 푥 Γx ρ − 푥휌−1푒 휆 Γx ρ 푎−1 1 − λ λ Γ ρ 푓 푥; 푎, 푏, ρ, λ = ______(7.8) 퐵 푎, 푏 Γ ρ aλρ

푎, 푏, ρ, λ, x > 0

Equation 7.8 is the beta-gamma distribution introduced by Kong et al. (2007). They also derived some properties of the limit of the density function and hazard function.

7.3.6 Beta-Gumbel Distribution – (Nadarajah and Kotz (2004))

Nadarajah and Kotz (2004) introduced the beta-Gumbel distribution, based on a composition of the classical beta distribution and the Gumbel distribution. They calculated expressions for the rth moment, gave some particular cases, and studied the density function.

190

The beta-Gumbel distribution is obtained as follows:

Let x − μ 퐺 푥 = exp − exp − ς be the cdf of Gumbel distribution and

1 x − μ x − μ 푔 푥 = exp − exp − exp − ς ς ς 푢 = exp −푢 ς is the pdf of the gumbel distribution. Where x − μ 푢 = exp − ς By using equation 7.1, the density function of the beta-gumbel distribution is given by

1 푢 푓 푥; 푎, 푏, 푢, ς = [exp −푢 ]a−1 [1 − exp −푢 ]b−1. exp −푢 퐵(푎, 푏) ς 푢 = e−푎푢 1 − e−푢 b−1______(7.9) 휍퐵 푎, 푏

푎, 푏, 푢, ς, 푥 > 0

Equation 7.9 is the beta-gumbel distribution introduced by Nadarajah and Kotz (2004). They calculated expressions for the rth moments, gave some particular cases and studied the density function.

7.3.7 Beta-Frèchet Distribution (Nadarajah and Gupta (2004))

Nadarajah and Gupta (2004) introduced the beta- Frèchet distribution, based on a composition of the classical beta distribution and the Frèchet distribution. They derived in their paper some of its properties and hazard function and of the hazard functions. They also gave an expression for the rth moment. Barreto-Souza et.al (2008) gave another expression for the moments and derived some additional mathematical properties.

The beta- Frèchet distribution is obtained as follows:

191

Let

x −λ 퐺 푥 = exp − ς be the cdf of standard Frèchet distribution and

λ휍λ x −λ 푔 푥 = exp − 푥1+λ ς be the pdf of the Frèchet distribution.

By using equation 7.1, the density function of the beta-Frèchet distribution is given by

b−1 x −λ x −λ λ휍λ exp −a 1 − exp − ς ς 푓 푥; 푎, 푏, μ, ς = ______(7.10) 푥1+λ 퐵 푎, 푏

푎, 푏, λ, ς, x > 0

Equation 7.10 is the beta-Frèchet distribution introduced by Nadarajah and Gupta (2004). They derived some of its properties and gave an expression for the rth moment. Barreto- Souza et.al (2008) gave another expression for the moments and derived some additional mathematical properties.

7.3.8 Beta-Maxwell Distribution (Amusan (2010))

Amusan (2010) introduced the beta-Maxwell distribution, based on a composition of the classical beta distribution and the Maxwell distribution. She defined and studied the three-parameter Maxwell distribution. She also discussed various properties of the distribution.

The beta-maxwell distribution is obtained as follows:

Let

3 푥2 2γ 2 , 2 퐺 푥 = π be the cdf of maxwell distribution and

192

푥 2 − 2 푥2푒 2∝2 푔 푥 = π α3 be the pdf of the maxwell distribution.

By using equation 7.1, the density function of the beta-maxwell distribution is given by

2 a−1 2 b−1 3 푥 3 푥 푥 2 2γ , 2γ , − 2 2α 2 2α 2 푥2푒 2∝2 1 − π π π α3 푓 푥; 푎, 푏, α = 퐵 푎, 푏

푥 2 a−1 b−1 − 1 2 3 푥2 2 3 푥2 2 푥2푒 2∝2 = γ , 1 − γ , ______(7.11) 퐵 푎, 푏 π 2 2α π 2 2α π α3

푓표푟 푎, 푏, α, 푥 > 0 푎푛푑 γ 푎, 푏 is an incomplete gamma function.

Equation 7.11 is the beta-Maxwell distribution introduced by Amusan (2010).

7.3.9 Beta-Pareto Distribution (Akinsete et.al (2008))

Akinsete et.al (2008) introduced the beta-Pareto distribution, based on a composition of the classical beta distribution and the Pareto distribution. They defined and studied properties of the four-parameter beta-pareto distribution.

The beta-pareto distribution is obtained as follows:

Let

푥 −k 퐺 푥 = 1 − 휃 be the cdf of pareto distribution and

푘휃푘 푔 푥 = 푥푘+1 be the pdf of the pareto distribution.

By using equation 7.1, the density function of the beta-pareto distribution is given by

193

a−1 b−1 푥 −k 푥 −k 푘휃푘 1 − 1 − 1 − 휃 휃 푥푘+1 푓 푥; 푎, 푏, 푘, 휃 = 퐵 푎, 푏

푘 푥 −k a−1 푥 −푘푏 −1 = 1 − ______(7.12) 휃퐵 푎, 푏 휃 휃

푎, 푏, 푘, 휃, 푥 > 0

Equation 7.12 is the beta-pareto distribution defined by Akinsete et.al.

7.3.10 Beta-Rayleigh Distribution (Akinsete and Lowe (2009))

Akinsete and Lowe (2009) introduced the beta-Rayleigh distribution, based on a composition of the classical beta distribution and the Rayleigh distribution.

The beta-Rayleigh distribution is obtained as follows:

Let

푥 2 − 퐺 푥 = 1 − 푒 2α2 be the cdf of Rayleigh distribution and

푥 2 푥 − 푔 푥 = 푒 α 2 α2 be the pdf of the Rayleigh distribution.

By using equation 7.1, the density function of the beta-Rayleigh distribution is given by

a−1 b−1 2 푥 2 푥 2 푥 − − 푥 − 1 − 푒 2α2 1 − 1 − 푒 2α2 푒 α 2 α2 푓 푥; 푎, 푏, 훼 = 퐵 푎, 푏

a−1 2 푥 2 푥 푥 − −푏 = 1 − 푒 2α2 푒 α 2 ______(7.13) α2퐵 푎, 푏

푎, 푏, 푘, 훼 > 0

194

Equation 7.13 is the beta-Rayleigh distribution. If 푎 = 푏 = 1, equation 7.13 reduces to Rayleigh distribution with parameter 훼. When 푎 = 1, the beta-rayleigh distribution reduces to the Rayleigh distribution with parameter 푘 = 훼 . 푏

7.3.11 Beta-generalized-logistic of type IV distribution (Morais (2009))

Morais (2009) introduced the beta-generalized logistic of type IV distribution, based on a composition of the classical beta distribution and the generalized logistic of type IV distribution. She discussed some of its special cases that belong to the beta-G such as the beta –generalized logistic of types I, II, and III and some related distributions like the beta-beta prime and the beta-F. The generalized logistic of type IV distribution was proposed by Prentice (1976) as an alternative to modeling binary response data with the usual symmetric logistic distribution.

The beta-generalized logistic of type IV distribution is obtained as follows:

Let

B 1 (p, q) −x 퐺 푥 = 1+e B(p, q) be the cdf of the generalized logistic of type IV distribution given by a normalized incomplete beta function and

e−qx 푔 푥 = B p, q (1 + e−x )p+q be the pdf of the generalized logistic of type IV distribution.

By using equation 7.1, the density function of the beta-generalized logistic of type IV distribution is given by

푓 푥; 푎, 푏, p, q B a, b 1−a−b eqx a−1 −x b−1 = −x p+q [B 1 (p, q)] [B e (q, p)] ______(7.14) B a, b 1 − e 1+e−x 1+e−x

푎, 푏, p, q, > 0, 푥 > 푅

195

Equation 7.14 is the beta-generalized logistic of type IV distribution introduced by Morais (2009).

7.3.12 Beta-generalized-logistic of type I distribution (Morais (2009))

Morais (2009) introduced the beta-generalized logistic of type I distribution which is a special case of the beta-generalized logistic of type IV distribution with q = 1.

The density function of the beta-generalized logistic of type I distribution is given by

pe−x[ 1 + e−ax p − 1]b−1 푓 푥; 푎, 푏, p = , ______(7.15) B a, b 1 + e−x a+pb

푎, 푏, p > 0, 푥 > 푅

Equation 7.15 is the beta-generalized logistic of type I distribution introduced by Morais (2009).

7.3.13 Beta-generalized-logistic of type II distribution (Morais (2009))

Morais (2009) introduced the beta-generalized logistic of type II distribution which is a special case of the beta-generalized logistic of type IV distribution with p = 1. This distribution can also be obtained through the transformation 푋 = −푌 where Y follows the beta-generalized logistic of type I distribution.

The density function of the beta-generalized logistic of type II distribution is given by

qe−bqx e−qx 푓 푥; 푎, 푏, q = 1 − ______(7.16) B a, b 1 + e−x qb +1 B a, b 1 + e−x q

푎, 푏, q, > 0, 푥 > 푅

Equation 7.16 is the beta-generalized logistic of type II distribution introduced by Morais (2009).

196

7.3.14 Beta-generalized-logistic of type III distribution (Morais (2009))

Morais (2009) introduced the beta-generalized logistic of type III distribution which is a special case of the beta-generalized logistic of type IV distribution with p = q. This distribution is symmetric when a=b.

The density function of the beta-generalized logistic of type III distribution is given by

1−a−b −qx B q, q e a−1 b−1 −x 푓 푥; 푎, 푏, q = −x 2q [B 1 (q, q)] [B e (q, q)] ___(7.17) B a, b 1 + e 1+e−x 1+e−x

푎, 푏, q, > 0, 푥 > 푅

Equation 7.17 is the beta-generalized logistic of type III distribution introduced by Morais (2009).

7.3.15 Beta-beta prime distribution (Morais (2009))

The beta-beta prime distribution can be obtained from the beta-generalized logistic of type IV distribution using the transformation 푥 = e−y where Y is a random variable following the beta-generalized logistic of type IV distribution.

The density function of the beta-beta prime distribution is given by

퐵 푝, 푞 1−푎−푏 푥푞−1 푥 푎−1 푏−1 푓 푥; 푎, 푏, 푝, q = 푝+푞 [퐵 (푞, 푝)] [퐵 1 (푝, 푞)] ____(7.18) 퐵 푎, 푏 1 + 푥 1+푥 1+푥

푎, 푏, 푞, > 0, 푥 > 푅

Equation 7.18 is the beta-beta prime distribution introduced by Morais (2009).

197

7.3.16 Beta-F distribution (Morais (2009))

The beta-F distribution can be obtained as follows:

Let

1 푣 푢 v 퐺 푥 = B x , 푢 푣 u 2 2 B , v 2 2 1+ x u be the cdf of the F distribution and

q v v 2 −1 x2 u 푔 푥 = 푢 푣 (u+v)/2 B , v 2 2 1 + u x be the pdf of the F distribution.

By using equation 7.1, the density function of the beta-F distribution is given by

푓 푥; 푎, 푏, u, v v a−1 v b−1 v 2 −1 −1 2 B a, b u x v u v u v = (u+v)/2 I x , I 1 , __(7.19) u v v u 2 2 v 2 2 B , 1 + x v 1+ x 2 2 1+ x u u u

푎, 푏, u, v, x > 0

Equation 7.19 is the beta-F distribution. It can easily be verified that if Y follows the beta generalized logistic of type IV distribution with parameters 푎, 푏, u, v then F = v e−y u follows the beta F distribution with parameters 푎, 푏, 2u, 2v,.

7.3.17 Beta-Burr XII distribution (Paranaíba et.al (2010))

Paranaíba et.al (2010) introduced the beta-burr XII distribution, based on a composition of the classical beta distribution and the burr XII distribution. They defined and studied the five parameter beta-burr XII distribution. They also derived some of its properties like moment generating function, mean and proposed methods of estimating its parameters.

198

The beta- burr XII distribution is obtained as follows:

Let

푥 p −푘 퐺 푥 = 1 − 1 + q be the cdf of burr XII distribution and

푥 p −푘−1 푔 푥 = 푝푘푞−푝 1 + 푥푝−1 q be the pdf of the burr XII distribution.

By using equation 7.1, the density function of the beta- burr XII distribution is given by

푓 푥; 푎, 푏, 푝, 푞, 푘 a−1 b−1 p −푘 p −푘 p −푘−1 푥 푥 −푝 푥 푝−1 1 − 1 + q 1 + q 푝푘푞 1 + q 푥 = 퐵 푎, 푏

a−1 푝푘푞−푝 푥푝−1 푥 p −푘 푥 p −푘푏 −1 = 1 − 1 + 1 + ______(7.20) 퐵 푎, 푏 q q for 푎, 푏, 푝, 푞, 푘, 푥 > 0.

Equation 7.20 is the beta-burr XII distribution introduced by Paranaíba et.al (2010). The beta-burr XII distribution contains well-known distributions as special sub-models.

For 푎 = 푏 = 1, equation 7.20 gives the burr XII distribution

푥 p −푘−1 푓 푥; 푝, 푞, 푘 = 푝푘푞−푘 푥푝−1 1 + q

For 푏 = 1, equation 7.20 gives the exponentiated burr XII distribution

a−1 푥 p −푘 푥 p −푘−1 푓 푥; 푎, 푝, 푞, 푘 = 푎푝푘푞−푘 푥푝−1 1 − 1 + 1 + q q

For 푎 = 푏 = 1, 푞 = 휆−1 and 푘 = 1 equation 7.20 gives the log logistic distribution

푓 푥; 푝, 푞, = 푝휆푝 푥푝−1 1 + 푥휆 p −2

199

For 푎 = 푝 = 1 equation 7.20 gives the beta-pareto type II

−푘푏 −1 푏푘 푥 푓 푥; 푏, 푘, 푞, = 1 + 푞 q

For 푎 = 푏 = 푝 = 1 equation 7.20 gives the beta-pareto type II

−푘−1 푘 푥 푓 푥; 푘, 푞, = 1 + 푞 q

7.3.18 Beta-Dagum distribution (Condino and Domma (2010))

Condino and Domma (2010) introduced the beta-dagum distribution, based on a composition of the classical beta distribution and the dagum distribution. They defined and studied its properties.

The beta- dagum distribution is obtained as follows:

Let

퐺 푥 = 1 + 푞푥−푝 −푎 be the cdf of dagum distribution and

푝 푞푎푥−푝−1 푔 푥 = (1 + 푞푥−푝 )1+a be the pdf of the dagum distribution.

By using equation 7.1, the density function of the beta- dagum distribution is given by

1 푝 푞푎푥−푝−1 푓 푥 = 1 + 푞푥−푝 −푎 a−1 1 − 1 + 푞푥−푝 −푎 b−1 ______(7.21) 퐵 푎, 푏 (1 + 푞푥−푝 )1+a for 푎, 푏, 푝, 푞, 푥 > 0.

Equation 7.21 is the beta-dagum distribution introduced by Condino and Domma (2010). The beta-dagum distribution contains the beta-fisk (log-logistic) distribution when

푎 = 1 and 푞 = 1 and the dagum distribution when 푎 = 푏 = 1. 휆

200

7.3.19 Beta-(fisk) log logistic distribution

This is a new distribution referred to by Paranaíba et.al (2010) as a special sub-model of beta-burr XII distribution. It can be constructed based on a composition of the classical beta distribution and the log logistic (fisk) distribution as follows.

Let

휆푥 p 퐺 푥 = 1 + 휆푥 p be the cdf of log logistic distribution and

휆푝 휆푥 p−1 푔 푥 = 1 + 휆푥 p 2 be the pdf of the log logistic distribution.

By using equation 7.1, the density function of the beta- log logistic distribution is given by

휆푥 p a−1 휆푥 p b−1 휆푝 휆푥 p−1 1 − 1 + 휆푥 p 1 + 휆푥 p 1 + 휆푥 p 2 푓 푥; 푎, 푏, 푝, 휆 = 퐵 푎, 푏

휆푝 휆푥 ap −1 = ______(7.22) 퐵 푎, 푏 1 + 휆푥 p a+b for 푎, 푏, 푝, 휆, 푥 > 0.

Equation 7.22 is the beta-log logistic distribution referred to by Paranaíba et.al (2010).

7.3.20 Beta-generalized half normal Distribution (Pescim, R.R, et.al (2009))

Pescim, R.R, et.al (2009) introduced the beta-generalized half normal distribution, based on a composition of the classical beta distribution and the generalized half normal distribution. They derived expansions for the cumulative distribution and density functions. They also obtained formal expressions for the moments of the four-parameter beta- generalized half normal distribution.

201

The beta- generalized half normal distribution is obtained as follows:

Let

푥 훼 푥 α 퐺 푥 = 2휙 − 1 = erf 휍 휍 2

Where

1 푥 2 푥 휙 푥 = 1 + erf and erf 푥 = 푒푥푝 −푡2 푑푡 2 2 휋 0 be the cdf of generalized half normal distribution and

α 1 푥 ퟐ휶 2 α 푥 − 푔 푥 = 푒 2 휍 , 0 ≤ 푥 π 푥 휍 be the pdf of the generalized half normal distribution.

By using equation 7.1, the density function of the beta-generalized half normal distribution is given by

α a−1 α b−1 α 1 푥 ퟐ휶 푥 푥 2 α 푥 − 2 휍 2휙 휍 − 1 2 − 2휙 휍 π 푥 휍 푒 푓 푥; 푎, 푏, α, 휍 = 퐵 푎, 푏

푎, 푏, α, 휍 > 0 ______(7.23)

Equation 7.23 is the beta- generalized half normal distribution defined by Pescim, R.R, et.al. (2009) It contains as special sub-models well known distributions. for α = 1, it reduces to beta half normal distribution given as

1 푥 ퟐ − a−1 b−1 2 휍 푥 푥 2 푒 2휙 휍 − 1 2 − 2휙 휍 π 휍 푓 푥; 푎, 푏, 휍 = 퐵 푎, 푏

푎, 푏, 휍 > 0 for a = b = 1, it reduces to generalized half normal distribution given by

202

α 1 푥 ퟐ휶 2 α 푥 − 푓 푥; α, 휍 = 푒 2 휍 π 푥 휍

α, 휍 > 0 further if α = 1, the generalized half normal distribution gives the half normal distribution

1 푥 ퟐ − 2 푒 2 휍 푓 푥; α, 휍 = π 휍

α, 휍 > 0 for 푏 = 1, it leads to the exponentiated generalized half normal distribution given as

α 1 푥 ퟐ휶 α a−1 2 α 푥 − 푥 푓(푥; 푎, 훼, 휍) = 푎 푒 2 휍 2휙 − 1 π 푥 휍 휍

7.4 Beta Exponentiated Generated Distributions Beta exponentiated generated distributions are based on the power of the cdfs.

7.4.1 Exponentiated generated distribution

Let

퐹 푥 = 퐺 푥 훼

Where 퐺 푥 is the old or parent cdf and 퐹 푥 is the new cdf

The new cdf in terms of the old is therefore given by

푓(푥) = 훼 퐺 푥 훼−1푔(푥)

The following are some of the examples of the exponentiated exponential distributions

203

7.4.2 Exponentiated exponential distribution (Gupta and Kundu (1999))

Let

푥 1 − 푔 푥 = 푒 훽 , β > 0, 0 < 푥 < ∞ β

푥 −( ) 퐺 푥 = 1 − 푒 훽

푥 훼 − 퐹 푥 = 1 − 푒 훽 and

푥 훼−1 푥 훼 − − 푓 푥 = 1 − 푒 훽 푒 훽 ______(7.24) β

Equation 7.24 is an example of the exponentiated generated distribution known as the exponentiated exponential distribution which has been extensively covered by Gupta and Kundu (1999).

7.4.3 Exponentiated Weibull distribution (Mudholkar et.al. (1995)) Let

푥 푝 푝 − 푔 푥 = 푥푝−1푒 훽 βp

푥 푝 − 퐺 푥 = 1 − 푒 훽

훼 푥 푝 − 퐹 푥 = 1 − 푒 훽

훼−1 푥 푝 푥 푝 훼 − − 푓 푥 = 1 − 푒 훽 푒 훽 ______(7.25) 훽푝

Equation 7.25 is the exponentiated Weibull distribution proposed by Mudholkar et.al. (1995).

204

7.4.4 Exponentiated Pareto distribution (Gupta et.al (1998)) Let

푥 −푝 푏−1 푔 푥 = 푝qp 푏푥−푝−1 1 − 푞

푥 −푝 푏 퐺 푥 = 1 − 푞

훼 푥 −푝 푏 퐹 푥 = 1 − 푞

훼−1 푥 −푝 푏 푝푏 푓 푥 = 훼 1 − 푥−푝푏 −1 ______(7.26) 푞 푞

Equation 7.26 is the exponentiated Pareto distribution.

7.4.5 Beta exponentiated generated distribution

푔 푥 훼 퐺 푥 훼−1 퐺 푥 훼 푎−1 1 − 퐺 푥 훼 푏−1 푓 푥 = 퐵 푎, 푏

푔 푥 훼 퐺 푥 푎훼 −1 1 − 퐺 푥 훼 푏−1 = , 푎 > 0, 훼 > 0, 푏 > 0, 퐵(푎, 푏)

−∞ < 푥 < ∞, 0 < 퐺 푥 훼 < 1 is the exponentiated generator distribution or exponentiated generated distribution. Below are some of the examples on exponentiated generated distributions.

7.4.6 Generalized beta generated distributions

To obtain generalized beta distributions, we simply replace 퐺 푥 by [퐺 푥 ]푝 in the beta generated distribution. Some of the many distributions which can arise within the generalized beta generated class of distributions can be derived from an arbitrary parent cumulative distribution function (cdf) 퐺 푥 , and the probability density function (pdf) 푓 푥 of the new class of generalized beta distributions is defined as

205

푝 푏−1 푝 푔(푥)[퐺(푥)]푎푝−1 1 − 퐺 푥 푓(퐺 푥); 푎, 푏 ) = ______(7.27) 퐵 푎, 푏 where 푎 > 0, 푏 > 0 and 푝 > 0 are additional shape parameters which aim to introduce skewness and to vary tail weight and 푔 푥 = 푑퐺(푥)/푑푥. the new class of generalized beta distributions generated by equation 7.27 includes as special sub-models the beta generated distributions discussed in section 7.2 above. Eugene et.al (2002) introduced the beta generated distributions for 푝 = 1 i.e 푏−1 푔(푥)[퐺(푥)]푎−1 1 − 퐺 푥 푓(푥) = 퐵 푎, 푏 and Cordeiro and Castro (2010) introduced Kumaraswamy (Kw) generated distributions for 푎 = 1 i. e

푝 푏−1 푓(푥) = 푏 푝 푔(푥)[퐺(푥)]푝−1 1 − 퐺 푥 Alternatively, the generalized beta generated distribution in equation 7.27 can be derived from the generalized beta distribution of the first kind in equation 5.1 as follows: Letting 푞 = 1 푦 푝 푏−1 푝 푦푝푎−1 1 − . 푔 푥 푔 푦 = 1 , 0 < 푦푝 < 1, 푎, 푏, 푝 > 0 1푎푝 퐵 푎, 푏 Now consider 퐺(푥) 푝 푦푝푎−1 1 − 푦푝 푏−1 퐹(푥) = 푑푦 0 퐵 푎, 푏 푎푝−1 푝 푏−1 푝 푔(푥) 퐺 푥 1 − 퐺 푥 ∴ 푓(푥) = , −∞ < 푥 < ∞, 푎, 푏, 푝 > 0 퐵 푎, 푏

7.4.7 Generalized beta exponential distribution (Barreto-Souza et al. (2010))

Let

퐺 푥 = 1 − exp(−λx) be the cdf of exponential distribution with parameter λ, and

푔 푥 = λ exp(−λx)

206

be the pdf of exponential distribution.

By using equation 7.27, the density function of the generalized beta-exponential distribution is given by

푓 푥; 푎, 푏, 푝, λ 푝 λ exp(−λx)[1 − exp(−λx) ]푝푎 −1 1 − 1 − exp −λx 푝 푏−1 = ______(7.28) 퐵 푎, 푏

Equation 7.28 is the generalized beta-exponential distribution.

7.4.8 Generalized beta-Weibull distribution Let

x c − 퐺 푥 = 1 − e β be the cdf of Weibull distribution and

푥 푐 푐 − 푔 푥 = 푥푐−1 푒 훽 β푐 be the pdf of Weibull distribution.

By using equation 7.27, the density function of the generalized beta-weibull distribution is given by

푓 푥; 푝, 푐, 푎, 푏, 훽

푝 푏−1 푥 푐 x c x c 푐 − − − 푝 푥푐−1 푒 훽 [1 − e β ]푝푎 −1 1 − 1 − e β β푐 = ___(7.29) 퐵 푎, 푏

Equation 7.29 is the generalized beta-Weibull distribution.

207

7.4.9 Generalized beta-hyperbolic secant distribution

Let

2 푥−휇 퐺 푥 = arctan 푒 휍 π be the cdf of the hyperbolic secant distribution and

1 푔 푥 = 푥 − 휇 ςπCosh 휍 be the pdf of the hyperbolic secant distribution

By using equation 7.27, the density function of the generalized beta-hyperbolic secant is given by

2 푥−휇 푎푝 −1 2 푥−휇 푝 푏−1 휍 휍 푝 π arctan 푒 1 − π arctan 푒 푓 푥; 푎, 푏, μ, ς, π = 푥 − 휇 ___(7.30)

퐵 푎, 푏 ςπ cosh 휍

푎 > 0, 푏 > 0, μ > 0, 휍 > 0 and 푥 ∈ ℝ

Equation 7.30 is the generalized beta-hyperbolic secant distribution. If μ = 0 and ς = 1, the generalized beta-hyperbolic secant distribution reduces to the standard beta- hyperbolic distribution in equation 7.7.

7.4.10 Generalized beta-normal distribution

The generalized beta-normal distribution is obtained as follows: Let 푥 − 휇 퐺 푥 = Φ 휍 be the cumulative density function of normal distribution with parameters 휇 and 휍, and 푥 − 휇 푔 푥 = 휙 휍 be the probability density function of the normal distribution.

208

By using equation 7.27, the density function of the generalized beta-normal distribution is given by

푥 − 휇 푥 − 휇 p 푏−1 푝휍−1[Φ ]푎푝−1 1 − Φ 푥 − 휇 푓 푥; 푎, 푏, 휇, 휍, 푝 = 휍 휍 휙 ___(7.31) 퐵 푎, 푏 휍 Where 푎 > 0, 푏 > 0, 휍 > 0, 푝 > 0 휇 ∈ ℝ 푎푛푑 푥 ∈ ℝ

Equation 7.31 is the generalized beta-normal distribution. If μ = 0 , ς = 1 and 푝 = 1 , the generalized beta-normal distribution reduces to the standard beta-normal distribution proposed by Eugene et.al (2002). If μ = 0, ς = 1, 푏 = 1 and 푎푐 = 2 the generalized beta-normal distribution coincides with the skew-normal distribution.

7.4.11 Generalized beta-log normal distribution

The generalized beta- log normal distribution is obtained as follows: Let 퐺 푥 = Φ log 푥 be the cumulative density function of standard log normal distribution, and 휙 log 푥 푔 푥 = 푥 be the probability density function of the standard log normal distribution. By using equation 7.27, the density function of the generalized beta-log normal distribution is given by

푝휙 log 푥 [Φ log푥 ]푎푝−1 1 − Φ log 푥 푝 푏−1 푓 푥; 푎, 푏, 푝 = ______(7.32) 푥퐵 푎, 푏 Where 푎 > 0, 푏 > 0, 푝 > 0 and 푥 ∈ ℝ

Equation 7.32 is the generalized beta-log normal distribution.

209

7.4.12 Generalized beta-Gamma distribution

The generalized beta-gamma distribution is obtained as follows: Let

Γx ρ 퐺 푥 = λ Γ ρ

be the cumulative density function of gamma distribution, and

ρ−1 x x − 푔 푥 = e λ λ be the probability density function of the gamma distribution. By using equation 7.27, the density function of the generalized beta-gamma distribution is given by

푎푝−1 푝 푏−1 ρ−1 x Γx ρ Γx ρ x − e λ λ 1 − λ λ Γ ρ Γ ρ 푓 푥; 푎, 푏, 푝, ρ, λ = ___(7.33) 퐵 푎, 푏 Where 푎 > 0, 푏 > 0, 푝 > 0, ρ > 0, 휆 > 0 and 푥 ∈ ℝ

Equation 7.33 is the generalized beta-gamma distribution.

7.4.13 Generalized beta-Frèchet distribution

The generalized beta- Frèchet distribution is obtained as follows: Let

x −λ 퐺 푥 = exp − ς

be the cumulative density function of Frèchet distribution, and λ x −λ 푔 푥 = exp − 푥휆−1 ς ς

210

be the probability density function of the Frèchet distribution. By using equation 7.27, the density function of the generalized beta-Frèchet distribution is given by

푓 푥; 푎, 푏, 푝, ρ, λ

푎푝−1 푝 푏−1 λ x −λ x −λ x −λ 휆−1 푝 ς exp − ς 푥 exp − ς 1 − exp − ς = ___(7.34) 퐵 푎, 푏 Where 푎 > 0, 푏 > 0, 푝 > 0, ρ > 0, 휆 > 0 and 푥 ∈ ℝ

Equation 7.34 is the generalized beta-Frèchet distribution.

211

8 Chapter VIII: Generalized Beta Distributions Based on Special Functions

8.1 Introduction

Beta, F, Gamma functions are among the many special functions. Most of these special functions can be expressed as hypergeometric function. This chapter looks at the generalized beta distributions based on special functions such as the hypergeometric distribution function, the confluent hypergeometric function, gauss Hypergeometric function among others.

8.2 Confluent Hypergeometric Function The confluent hypergeometric (Kummer‘s) function is denoted by the symbol

1퐹1 푎; 푐; 푥 and represent the series 푎 푥 푎 푎 + 1 푥2 푎 푎 + 1 (푎 + 2) 푥3 퐹 푎; 푐, 푥 = 1 + + + + ⋯ 1 1 푐 1! 푐 푐 + 1 2! 푐 푐 + 1 (푐 + 2) 3! ∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛 − 1) 푥푛 = 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 1) 푛! n=0 for 푐 ≠ 0, −1, −2, … It can simply be expressed in the following form ∞ 푛 푎 n 푥 1퐹1 푎; 푐, 푥 = , c ≠ 0, −1, −2, … 푐 푛 푛! n=0

where 푎 n is the Pochhammer‘s symbol defined by

푎 n = a a + 1 … a + n − 1 Γ(a + 1) = ; n is a positive integer Γ(a)

푎 0 = 1 If a is a non-positive integer, the series terminates. The integral representation is given as

1 Γ(c) 푎−1 푐−푎−1 푥푢 1퐹1 푎; 푐, 푥 = 푢 1 − 푢 푒 푑푢 ______(8.1) Γ(a)Γ(c − a) 0

212

where 푐 > 푎 > 0 We verify that equation 8.1 is a distribution ∞ 1 1 (푥푢)푛 푢푎−1 1 − 푢 푐−푎−1푒푥푢 푑푢 = 푢푎−1 1 − 푢 푐−푎−1 푑푢 푛! 0 0 푛=0 ∞ 1 1 = 푥푛 푢푛+푎−1 1 − 푢 푐−푎−1 푑푢 푛! 푛=0 0 ∞ 푥푛 1 = 푢푛+푎−1 1 − 푢 푐−푎−1 푑푢 푛! 푛=0 0 ∞ 푥푛 = B(n + a, c − a) 푛! 푛=0 ∞ 1 푥푛 Γ(n + a)Γ(c − a) ⟹ 푢푎−1 1 − 푢 푐−푎−1푒푥푢 푑푢 = 푛! Γ(n + c) 0 푛=0 but Γ n + a = (n + a − 1)Γ n + a − 1 = a a + 1 a + 2 … n + a − 1 Γ(a) 1 a a + 1 a + 2 … n + a − 1 Γ(a)Γ(c − a) 푥푛 ⟹ 푢푎−1 1 − 푢 푐−푎−1푒푥푢 푑푢 = 0 c c + 1 c + 2 … n + c − 1 Γ(c) 푛!

= 1퐹1 푎; 푐, 푥 B(a, c − a) Thus 1 1 푎−1 푐−푎−1 푥푢 1퐹1 푎; 푐, 푥 = 푢 1 − 푢 푒 푑푢 퐵(푎, 푐 − 푎) 0

8.3 The Differential Equations The first and second order differential equations can be expressed as follows;

푑 푎 1 푎 푎 + 1 2푥 푎 푎 + 1 (푎 + 2) 3푥 퐹 푎; 푐; 푥 = 0 + + + + ⋯ 푑푥 1 1 푐 1! 푐 푐 + 1 2! 푐 푐 + 1 (푐 + 2) 3! 푎 푎 푎 + 1 푥 푎 푎 + 1 (푎 + 2) 푥2 = + + + ⋯ 푐 푐 푐 + 1 1! 푐 푐 + 1 (푐 + 2) 2! 푎 푎 + 1 푥 푎 + 1 푎 + 2 푥2 = 1 + + + ⋯ 푐 푐 + 1 1! 푐 + 1 푐 + 2 2!

213

푎 = 퐹 푎 + 1; 푐 + 1; 푥 푐 1 1

푑2 푎 푎 + 1 푎 + 1 푎 + 2 푥 퐹 푎; 푐; 푥 = 0 + + + ⋯ 푑푥2 1 1 푐 푐 + 1 푐 + 1 푐 + 2 1! 푎 푎 + 1 푎 + 2 푥 = 1 + + ⋯ 푐 푐 + 1 푐 + 2 1! 푎 푎 + 1 = 퐹 푎 + 2; 푐 + 2; 푥 푐 푐 + 1 1 1 In applied mathematics, special functions are used in solving differential equation. Thus

1퐹1 푎; 푐; 푥 is a solution to a certain differential equation which is derived as follows:-

Using the operator 푑 훿 = 푥 푑푥 Therefore 푑 훿푥푛 = 푥 푥푛 = 푥푛푥푛−1 = 푛푥푛 푑푥 훿 + 푐 − 1 푥푛 = 훿푥푛 + (푐 − 1)푥푛 ∴ 훿 + 푐 − 1 푥푛 = (푛 + 푐 − 1)푥푛 훿 훿 + 푐 − 1 푥푛 = 훿(푛 + 푐 − 1)푥푛 = 훿푥푛 (푛 + 푐 − 1) = 푛푥푛 (푛 + 푐 − 1) ∴ 훿 훿 + 푐 − 1 푥푛 = 푛(푛 + 푐 − 1)푥푛 훿 + 푎 훿 + 푏 푥푛 = 훿 + 푎 훿푥푛 + 푏푥푛 = 훿 + 푎 푛푥푛 + 푏푥푛 = 훿 + 푎 푛푥푛 + 훿 + 푎 푏푥푛 = 훿푛푥푛 + 푎푛푥푛 + 훿푏푥푛 + 푎푏푥푛 = 푛2푥푛 + 푎푛푥푛 + 푏푛푥푛 + 푎푏푥푛 = (푛2 + 푎푛 + 푏푛 + 푎푏)푥푛 = (푛(푛 + 푎) + 푏(푛 + 푎))푥푛 ∴ 훿 + 푎 훿 + 푏 푥푛 = 푛 + 푎 (푛 + 푏) 푥푛

214

∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛 − 1) 푥푛 훿 훿 + 푐 − 1 퐹 푎; 푐; 푥 = 훿 훿 + 푐 − 1 1 1 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 1) 푛! n=0 ∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛 − 1) 푥푛 = 훿 훿 + 푐 − 1 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 1) 푛! n=0 ∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛 − 1) 푥푛 = 푛 푛 + 푐 − 1 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 1) 푛! n=0 ∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛 − 1) 푥푛 = 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 2) (푛 − 1)! n=0 ∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛 − 1) 푥푛−1 = 푥 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 2) (푛 − 1)! n=1 ∞ 푎 푎 + 1 (푎 + 2) … (푎 + 푛) 푥푛 = 푥 푐 푐 + 1 푐 + 2 … (푐 + 푛 − 1) 푛! n=1 by replacing n with n+1 ∞ 푎 푎 + 1 푎 + 2 … 푎 + 푛 − 1 푥푛 = 푥 (푎 + 푛) 푐 푐 + 1 푐 + 2 … 푐 + 푛 − 1 푛! n=0

∞ 푎 푎 + 1 푎 + 2 … 푎 + 푛 − 1 푥푛 = 푥 (푎 + 훿) 푐 푐 + 1 푐 + 2 … 푐 + 푛 − 1 푛! n=0

∞ 푎 푎 + 1 푎 + 2 … 푎 + 푛 − 1 푥푛 = 푥(푎 + 훿) 푐 푐 + 1 푐 + 2 … 푐 + 푛 − 1 푛! n=0

∴ 훿 훿 + 푐 − 1 1퐹1 푎; 푐; 푥 = 푥(푎 + 훿) 1퐹1 푎; 푐; 푥 i.e 훿 훿 + 푐 − 1 푦 = 푥(푎 + 훿)푦 where

푦 = 1퐹1 푎; 푐; 푥 ∴ [훿2 + 훿(푐 − 1)]푦 = 푥(푎 + 훿)푦 ⟹ 훿2푦 + [훿 푐 − 1 − 푥(푎 + 훿)]푦 = 0 ∴ 훿2푦 + 푐 − 1 훿푦 − 푥푎푦 − 푥훿푦 = 0

215

∴ 훿2푦 + 푐 − 1 − 푥 훿푦 − 푥푎푦 = 0 훿(훿푦) + 푐 − 1 − 푥 훿푦 − 푥푎푦 = 0 푑 푑 훿 푥 푦 + 푐 − 1 − 푥 푥 푦 − 푥푎푦 = 0 푑푥 푑푥 푑 푑 푑푦 푥 푥 푦 + (푐 − 1 − 푥)푥 − 푥푎푦 = 0 푑푥 푑푥 푑푥 푑 푑 푑푦 ∴ 푥 푦 + (푐 − 1 − 푥) − 푎푦 = 0 푑푥 푑푥 푑푥 푑2푦 푑푦 푑푦 푥 + + (푐 − 1 − 푥) − 푎푦 = 0 푑푥2 푑푥 푑푥 푑2푦 푑푦 푥 + (푐 − 푥) − 푎푦 = 0 푑푥2 푑푥 Which is the Kummer‘s differential equation. The confluent hypergeometric function satisfies this equation. The confluent hypergeometric distribution due to Gordy (1988a) is a special case of the confluent hypergeometric distribution given in equation 8.1 where 푐 = 푎 + 푏, 푢 = 푥 and 푥 = −훾 such that 1 Γ(a + b) 푎−1 푎+푏−푎−1 −훾푥 1퐹1 푎; 푎 + 푏, −훾 = 푥 (1 − 푥) 푒 푑푥 Γ(a)Γ(a + b − a) 0 1 Γ(a + b) 푎−1 푏−1 −훾푥 1퐹1 푎; 푎 + 푏, 푥 = 푥 (1 − 푥) 푒 푑푥 Γ(a)Γ(b) 0

1 푥푎−1(1 − 푥)푏−1푒−훾푥 푑푥 1 = 0 1퐹1 푎; 푎 + 푏, −훾 B(a, b)

푥푎−1[1 − 푥]푏−1푒 −훾푥 푓 푥 = , ______(8.2) 퐵 푎, 푏 1퐹1(푎; 푎 + 푏; −훾) for 0 < x < 1, a > 0, b > 0, −∞ < 훾 < ∞ and 1퐹1 is the confluent hypergeometric function. If 훾 = 0, then the moment generating function denoted by 퐵 푎 − 푡, 푏 퐹 푎 − 푡; 푎 + 푏 − 푡; −훾 푀 푡 = 1 1 ______(8.3) 퐵 푎, 푏 1퐹1 푎; 푎 + 푏; −훾 reduces to the standard beta pdf given by equation 2.1

216

8.4 Gauss Hypergeometric Function.

The Gaussian hypergeometric function denoted by the symbol 2퐹1 푎, 푏; 푐; 푥 represents the series 푎푏 푥 푎 푎 + 1 푏 푏 + 1 푥2 퐹 푎, 푏; 푐; 푥 = 1 + + + ⋯ 2 1 푐 1! 푐 푐 + 1 2! where 푐 ≠ 0, −1, −2, … It can simply be expressed in the following form; ∞ 푛 푎 n 푏 n푥 2퐹1 푎, 푏; 푐; 푥 = , c ≠ 0, −1, −2, … ______(8.4) 푐 푛 푛! n=0

where 푎 n is the Pochhammer‘s symbol defined by

푎 n = a a + 1 … a + r − 1 Γ(a + 1) = ; (n is a positive integer) Γ(a)

푎 0 = 1

If a is a non-positive integer, then 푎 n is zero for n > −푎, and the series terminates. The Euler‘s integral representation is given as

1 Γ(c) 푎−1 푐−푎−1 −푏 2퐹1 푎, 푏; 푐; 푥 = 푢 (1 − 푢) (1 − 푥푢) 푑푢 ______(8.5) Γ(a)Γ(c − a) 0

where 푐 > 푎 > 0

8.5 The Differential Equations The first and second order differentials can be expressed as follows;

푑 푎푏 푎 푎 + 1 푏 푏 + 1 퐹 푎, 푏; 푐; 푥 = + 푥 + ⋯ 푑푥 2 1 푐 푐 푐 + 1 푎푏 푎 + 1 푏 + 1 푎 + 1 푎 + 2 푏 + 1 (푏 + 2) 푥2 = 1 + 푥 + … 푐 푐 + 1 푐 + 1 (푐 + 2) 2! 푎푏 = 퐹 푎 + 1, 푏 + 1; 푐 + 1; 푥 푐 2 1

217

푑2 푎푏 푎 + 1 푏 + 1 푎 + 1 푎 + 2 푏 + 1 (푏 + 2) 퐹 푎, 푏; 푐; 푥 = + 푥 + ⋯ 푑푥2 2 1 푐 푐 + 1 푐 + 1 (푐 + 2) 푎푏 푎 + 1 푏 + 1 푎 + 2 (푏 + 2) = 1 + 푥 + ⋯ 푐 푐 + 1 (푐 + 2) 푎 푎 + 1 푏 푏 + 1 = 퐹 푎 + 2, 푏 + 2; 푐 + 2; 푥 푐 푐 + 1 2 1

We now wish to derive a differential equation whose solution is 2퐹1 푎, 푏; 푐; 푥 Let 푑 훿 = 푥 푑푥 then

훿 훿 + 푐 − 1 2퐹1 푎, 푏; 푐; 푥 ∞ 푎 푎 + 1 … 푎 + 푛 − 1 푏 푏 + 1 … 푏 + 푛 − 1 푥푛 = 훿 훿 + 푐 − 1 푐 푐 + 1 … (푐 + 푛 − 1) 푛! 푛=0 ∞ 푎 푎 + 1 … 푎 + 푛 − 1 푏 푏 + 1 … 푏 + 푛 − 1 푥푛 = 푐 푐 + 1 … (푐 + 푛 − 2) (푛 − 1)! 푛=0 ∞ 푎 푎 + 1 … 푎 + 푛 − 1 푏 푏 + 1 … 푏 + 푛 − 1 푥푛−1 = 푥 푐 푐 + 1 … (푐 + 푛 − 2) (푛 − 1)! 푛=1 replace n by n+1. Then

훿 훿 + 푐 − 1 2퐹1 푎, 푏; 푐; 푥 ∞ 푎 푎 + 1 … 푎 + 푛 − 1 푏 푏 + 1 … 푏 + 푛 − 1 푎 + 푛 (푏 + 푛) 푥푛 = 푥 푐 푐 + 1 … (푐 + 푛 − 1) 푛! 푛=1 ∞ 푎 푎 + 1 … 푎 + 푛 − 1 푏 푏 + 1 … 푏 + 푛 − 1 푎 + 훿 (푏 + 훿) 푥푛 = 푥 푐 푐 + 1 … (푐 + 푛 − 1) 푛! 푛=0 ∞ 푎 푎 + 1 … 푎 + 푛 − 1 푏 푏 + 1 … 푏 + 푛 − 1 푥푛 = 푥 푎 + 훿 (푏 + 훿) 푐 푐 + 1 … (푐 + 푛 − 1) 푛! 푛=0

∴ 훿 훿 + 푐 − 1 2퐹1 푎, 푏; 푐; 푥 = 푥 푎 + 훿 (푏 + 훿) 2퐹1 푎, 푏; 푐; 푥 훿 훿 + 푐 − 1 퐹 = 푥 푎 + 훿 (푏 + 훿) 퐹 where

218

푑 퐹 = 퐹 푎, 푏; 푐; 푥 and 훿 = 푥 2 1 푑푥 ∴ 훿 훿퐹 + 푐 − 1 퐹 = 푥 푎 + 훿 (푏퐹 + 훿퐹) 푑퐹 푑 ⟹ 훿 푥 + 푐 − 1 퐹 = 푥 푎 + 훿 푏퐹 + 푥 퐹 푑푥 푑푥 푑퐹 푑퐹 ⟹ 훿 푥 + 푐 − 1 훿퐹 = 푥 푎 + 훿 푏퐹 + 푥 푑푥 푑푥 푑 푑퐹 푑퐹 푑퐹 ⟹ 푥 푥 + 푐 − 1 푥 = 푥 푎 + 훿 푏퐹 + 푥 푑푥 푑푥 푑푥 푑푥 푑 푑퐹 푑퐹 푑퐹 ⟹ 푥 + 푐 − 1 = 푎 + 훿 푏퐹 + 푥 푑푥 푑푥 푑푥 푑푥 푑2퐹 푑퐹 푑퐹 푑퐹 푑퐹 푑퐹 푥 + + 푐 − = 푎푏퐹 + 푎푥 + 푏훿퐹 + 훿푥 푑푥2 푑푥 푑푥 푑푥 푑푥 푑푥

푑2퐹 푑퐹 푑퐹 푑퐹 푑퐹 푑퐹 ∴ 푥 + 푐 = 푎푏퐹 + 푎푥 + 푏푥 + 푥 푥 푑푥2 푑푥 푑푥 푑푥 푑푥 푑푥 푑퐹 푑2퐹 푑퐹 = 푎푏퐹 + (푎 + 푏)푥 + 푥 푥 + 푑푥 푑푥2 푑푥 푑2퐹 푑퐹 ∴ 푥 1 − 푥 + 푐 − 푎푥 − 푏푥 − 푥 − 푎푏퐹 = 0 푑푥2 푑푥 푑2퐹 푑퐹 ∴ 푥 1 − 푥 + 푐 − (푎 + 푏 + 1)푥 − 푎푏퐹 = 0 푑푥2 푑푥

This is the differential equation whose solution is 2퐹1 푎, 푏; 푐; 푥 i.e the Gaussian hypergeometric function satisfies this second order linear differential equation.

The Gauss hypergeometric distribution due to Armero and Bayarri (1994) is a special case of the Gaussian hypergeometric distribution given in equation 8.5 where 푐 = 푎 + 푏, 푥 = −푧 , 푢 = 푥, and −푏 = 훾 such that 1 Γ(a + b) 푎−1 푎+푏−푎−1 −푏 2퐹1 훾, 푏; 푎 + 푏; −푧 = 푥 (1 − 푥) (1 + 푧푥) 푑푥 Γ(a)Γ(a + b − a) 0

1 푥푎−1(1 − 푥)푏−1(1 + 푧푥)훾 푑푥 1 = 0 2퐹1 훾, 푏; 푎 + 푏; −푧 B(a, b)

219

푥푎−1(1 − 푥)푏−1/(1 + 푧푥)훾 푓 푥 = ______(8.6) 퐵 푎, 푏 2퐹1(훾, 푎; 푎 + 푏; −푧) for 0 < x < 1, a > 0, b > 0, and −∞ < 훾 < ∞. If 훾 = 0, then equation 8.6 reduces to beta of the first kind given by equation 2.1 Nadarajah and Kotz (2004) also studied the beta distribution based on the Gauss hypergeometric function. They defined the distribution as follows:

푏퐵 푎, 푏 푓 푥 = 푥푎+푏−1 퐹 (1 − 훾, 푎; 푎 + 푏; 푥), ______(8.7) 퐵 푎, 푏 + 훾 2 1

When 푏 = 0 in equation 8.7, then it reduces to the Classical beta distribution with parameters a and 훾.

8.6 Appell Function

Nadarajah and Kotz (2006) introduced a beta function based on the Appell function. The distribution is given by

퐶푥훼−1(1 − 푥)훽−1 푓 푥 = , ______(8.8) (1 − 푢푥)휌 (1 − 푣푥)휆 for 0 < 푥 < 1, 훼 > 0, 훽 > 0, 휌 > 0, 휆 > 0, −1 < 푢 < 1, −1 < 푣 < 1 where C denotes the normalizing constant given by 1 = 퐵 훼, 훽 퐹 (훼, 휌, 휆, 훼 + 훽; 푢, 푣) 퐶 1 and 퐹1 denotes the Appell function of the first kind. This distribution is flexible and contains several of the known generalizations of the Classical beta distribution as particular cases. The Classical beta distribution is the particular case for either 휌 = 0 and 휆 = 0 or 휌 = 0 and 푣 = 0 or 푢 = 0 and 휆 = 0 or 푢 = 0 and 푣 = 0.Libby and Novick‘s beta distribution in equation 4.1 is the particular case for either 휌 = 훼 + 훽 and 휆 = 0 or 푢 = 푣 and 휌 + 휆 = 훼 + 훽. Armero and Bayarri‘s Gauss hypergeometric distribution in equation 8.6 is the particular case for either 휌 = 0 or 휆 = 0

220

8.7 Nadarajah and Kotz’s Power Beta Distribution

The generalization due to Nadarajah and Kotz (2004) is based on the basic characterization that if X and Y are independent gamma distributed random variables with common scale parameter then the ratio X/(X+Y) is a beta distribution of the first kind. If one consider the distribution of 푊 = 푋푐 / (푋푐 + 푌푐 ) 퐶 > 0, referred to as power beta, then the pdf and cdf of W can be expressed as

1 푤− 1+푏/푐 1 − 푤 푏/푐−1 1 − 푤 푐 푓(푤) = 푏 퐹 푏, 푎 + 푏; 푏 + 1; − 푏푐퐵(푎, 푏) 2 1 푤

1 1 1 − 푤 푐 1 − 푤 푐 − 퐹 푏 + 1, 푎 + 푏 + 1; 푏 + 2; − 푤 2 1 푤 and

1 1 − 푤 푏/푐 1 − 푤 1/푐 퐹 푤 = 1 − 퐹 푏, 푎 + 푏; 푏 + 1; − . 푏퐵(푎, 푏) 푤 2 1 푤

Simpler expressions can be obtained for integer values of a, b and c.

8.8 Compound Confluent Hypergeometric Function The compound confluent hypergeometric distribution due to Gordy (1988b) is given by 푥푎−1[1 − 푣푥]푏−1{휃 + (1 − 휃)푣푥}−푟 푒 −푠푥 푓 푥 = 푠 푠 , ______(8.9) −푎 퐵 푎, 푏 푣 exp − 푣 Φ1(푏, 푟, 푎 + 푏, 푣 , 1 − 휃) for 0 < a < 1, b > 0, −∞ < 푟 < ∞, −∞ < 푠 < ∞, 0 ≤ 푣 ≤ 1, 휃 > 0 and Φ1 is the confluent hypergeometric function of two variables defined by

∞ ∞

훼 푚+푛 (훽)푛 푚 푛 Φ1 훼, 훽, 훾, 푥, 푦 = 푥 푦 훾 푚+푛 푚! 푛! 푚 =0 푛=0

221

When 푠 = 0, 푟 = 푎 + 푏, 푣 = 1−푐 and 휃 = 1 − 푐 in Equation 8.9, then it reduces to 푞 McDonald and Xu‘s five parameter generalized beta distribution in equation 6.1. When 푠 = 0, and 푣 = 1 in Equation 8.9, then it reduces to gauss hypergeometric distribution due to Armero and Bayarri (1994) in equation 8.6. When 푣 = 1, and 휃 = 1 in Equation 8.9, then it reduces to confluent hypergeometric distribution due to Gordy (1988a) in equation 8.2. When 푣 = 0, and > 0 , Equation 8.9 becomes the gamma distribution.

8.9 Beta-Bessel Distribution (A.K Gupta and S.Nadarajah (2006))

A.K Gupta and S. Nadarajah (2006) introduced three types of beta-bessel distributions, based on a composition of the classical beta distribution and the Bessel function. They derived expressions for their shapes, particular cases and the nth moments. They also discussed estimation by the method of maximum likelihood and Bayes estimation.

The beta- bessel distribution is obtained as follows:

The first generalization of equation 2.1 is given by the pdf

휶−1 휷−1 푓 푥 = C푥 1 − 푥 퐼푣 푐푥 ______(8.10)

For 0 < 푥 < 1, 푣 > 0, 훼 > 0, 훽 > 0 푎푛푑 푐 ≥ 0 , where C denotes the normalizing constant and can be determined as

1 푐푣훤 훼 + 푣 훤 훽 α + 푣 α + 푣 + 1 = 퐹 , ; 푣 푣 2 3 퐶 2 훤 훼 + 훽 + 푣 훤 푣 + 1 2 2 α + 푣 + β α + 푣 + β + 1 푐2 + 1, , ; 2 2 4

퐼푣 푥 is the modified Bessel function defined by

푥푣 1 퐼 푥 = 1 − 푡2 푣+1/2 exp 푥푡 푑푡 푣 푣 휋 2 푣 + 1/2 −1

The classical beta pdf (2.1) arises as particular case of 8.10 for 푣 = 0 and 푐 = 0.

The second generalization of equation 2.1 is given by the pdf

222

휶−1 휷−1 푓 푥 = C푥 1 − 푥 exp(푐푥)퐼푣 푐푥 ______(8.11)

For 0 < 푥 < 1, 푣 > 0, 훼 > 0, 훽 > 0 푎푛푑 푐 ≥ 0 , where C denotes the normalizing constant and can be determined as

1 푐푣훤 훼 + 푣 훤 훽 1 = 퐹 푣 + , 훼 + 푣; 2푣 + 1, 훼 + 훽 + 푣; 2푐 푣 2 2 퐶 2 훤 훼 + 훽 + 푣 훤 푣 + 1 2

The classical beta pdf (2.1) arises as the particular case of 8.11 for 푣 = 0 and 푐 = 0.

The third and final generalization of equation 2.1 is given by the pdf

휶−1 휷−1 푓 푥 = C푥 1 − 푥 exp(−푐푥)퐼푣 푐푥 ______(8.12)

For 0 < 푥 < 1, 푣 > 0, 훼 > 0, 훽 > 0 푎푛푑 푐 ≥ 0 , where C denotes the normalizing constant and can be determined as

1 푐푣훤 훼 + 푣 훤 훽 1 = 퐹 푣 + , 훼 + 푣; 2푣 + 1, 훼 + 훽 + 푣; 2푐 푣 2 2 퐶 2 훤 훼 + 훽 + 푣 훤 푣 + 1 2

The classical beta pdf (2.1) arises as the particular case of 8.12 for 푣 = 0 and 푐 = 0

223

9 Chapter IX: Summary and Conclusion

9.1 Summary

This project has looked at three methods of generalizing the classical beta distribution. The work could be summarized as follows:

In Chapter I, a general introduction covering the work on statistical distributions in general is given followed by a brief background on classical beta distribution. The limitation of the classical beta distribution is noted leading to the main objective of adding more parameters to make the distribution more flexible for analyzing empirical data. The recent works in literature is also highlighted. In Chapter II, the classical beta distribution is constructed from the beta function and the relationship between the beta function and the Gamma function is shown. Its construction from ratio transformation of two independent gamma variables is also done. Other constructions from the rth order statistic of the Uniform distribution and from the stochastic processes are also looked at. Some properties, shapes, and special cases of the classical beta are also covered under this chapter. In Chapter III, looks at the transformation of the classical beta distribution on the domain (0,1) to the extended beta distribution of the second kind on the domain (0, ∞). It also looked at some properties, shapes, and special cases of the beta distribution of the second kind. In Chapter IV, highlights some of the three parameter beta distributions used by various authors in literature. In particular, the well known Libby and Novick (1982) three parameter beta distribution and the McDonald (1995) three parameter beta distribution. In Chapter V, the four parameter beta distributions are constructed. Their properties and special cases are also given. In Chapter VI, covers the construction of the five parameter beta distributions due to McDonald (1995) giving some of its properties and special cases. It further extends the five parameter beta distribution due to McDonald (1995) by introducing a sixth parameter to give a five parameter weighted beta distribution. Some of the properties and special cases of this five parameter

224

weighted beta distribution have also been looked at. The chapter also looked at the construction of the five parameter exponential generalized beta distribution and its special cases. In Chapter VII, gives a new family of generalized beta distributions based on the work of Eugene et al. (2002) using the generator approach. It covers the beta generated distributions, exponentiated generated distributions and the generalized beta generated distributions. In Chapter VIII, the generalized beta distributions based on some special functions are covered. In particular, the Confluent Hypergeometric function, the Gauss Hypergeometric function, the Bessel and the Appell function. In Chapter IX, a summary and concluding remark of the work is given followed by references and an appendix.

9.2 Conclusion This project has looked at both the two kinds of two parameter beta distributions. It has reviewed various methods of constructing the beta distributions, their properties which include expressions th for the r moments, first four moments, mode, coefficient of variation, coefficient of skewness, coefficient of kurtosis and their shapes. It has also presented the special or limiting cases. However, there are other methods which can be explored in extending this work e.g. using mixtures of distributions. Other areas of research which need to be looked at are: further properties of the beta distributions, estimation and applications.

225

10 Appendix

A.1 Classical Beta Distribution

Pdf 푥푎−1(1 − 푥)푏−1 푓 푥 = , 푎 > 0, 푏 > 0, 퐵(푎, 푏) 0 < 푥 < 1 Cumulative distribution function 1 푥 퐹 푥 = 푡푎−1(1 − 푡)푏−1 푑푡 퐵(푎, 푏) 0 rth order moment Γ(a + r) Γ(a + b)

Γ(a + b + r) Γ(a) a Mean a + b

Mode (푎 − 1)

(푎 + 푏 − 2) Variance ab

a + b + 1 a + b 2 Skewness 2 b − a a + b + 1

a + b + 2 ab Kurtosis a + b + 1 3[a2(b + 2) − 2ab + b2(a + 2)]

푎푏 a + b + 2 a + b + 3

A.2 Power Distribution (b=1 in table A.1)

Pdf 푓 푥 = 푎푥푎−1, 푎 > 0, 0 < 푥 < 1

Cumulative distribution function 퐹(푥) = 푥푎 th 푎 r order moment a + r 푎 Mean a + 1 Mode 1 푎 Variance (a + 2)(a + 1)2 Skewness 2푎(1 − 푎) (a + 2)

(a + 3) a

Kurtosis 푎 + 2 3 3푎2 − 푎 + 2

푎 a + 3 a + 4

226

A.3 Uniform Distribution (a=b=1 in table A.1)

Pdf 푓 푥 = 1, 0 < 푥 < 1

rth order moment 1

1 + r Mean 1

Mode Any value in the interval [0,1]

Variance 1

12 Skewness 0

Kurtosis 1.8

ퟏ A.4 Arcsine Distribution (a=b= in table A.1) ퟐ

Pdf 1 푓 푥 = , 0 < 푥 < 1 휋 푥(1 − 푥) Cumulative distribution function 2 F x = arcsin 푥 휋 rth order moment 1 Γ + 푟 2 1

rΓ r Γ 2

Mean 1

2 Mode 1

2 Variance 0.125 Skewness 0

Kurtosis 1.5

227

A.5 Triangular Shaped (a=1, b=2 in table A.1)

Pdf 푓 푥 = 2 − 2푥, 0 < 푥 < 1

Cumulative distribution function 퐹 푥 = 2푥 − 푥2 rth order moment 2rΓ 푟

Γ r + 3

Mean 0.3333

Mode 0

Variance 0.0556

Skewness 0.5657

Kurtosis 2.4

A.6 Triangular Shaped (a=2, b=1 in table A.1)

Pdf 푓 푥 = 2푥, 0 < 푥 < 1 Cumulative distribution function 퐹 푥 = 푥2 rth order moment 2Γ 2 + 푟

Γ r + 3 Mean 0.6667 Mode 1 Variance 0.0556 Skewness -0.5657 Kurtosis 2.4

A.7 Parabolic Shaped (a=b=2 in table A.1)

Pdf 푓 푥 = 6푥 1 − 푥 , 0 < 푥 < 1 Cumulative distribution function 퐹 푥 = 3푥2 − 2푥3 rth order moment 6

r + 3 (r + 2) Mean 0.5 Mode 0.5 Variance 0.05 Skewness 0 Kurtosis 2.1429

228

ퟑ 퐲+퐑 A.8 Wigner Semicircle (a=b= , 퐚퐧퐝 퐱 = in table A.1) ퟐ ퟐ퐑

Pdf 2 푅2 − 푥2 푓 푥 = , −푅 < 푥 < 푅 푅2휋 rth order moment 푅 2푟 1 2푟

2 푟 + 1 푟

Mean 0

Mode 0

Variance 푅2

Skewness 0

Kurtosis 2

퐲 ퟏ A.9 Beta Distribution of the Second Kind (using 퐱 = or in table A.1) ퟏ+퐲 ퟏ+퐲

Pdf 푥푎−1 푓 푥 = , 푎 > 0, 푏 > 0, 0 < 푥 < ∞ 퐵 푎, 푏 1 + x a+b rth order Γ 푎 + 푟 Γ 푏 − 푟 moment Γ 푎 Γ 푏 푎 Mean 푏 − 1 Mode 푏 − 1

푎 + 1 Variance 푎(푎 + 푏 − 1)

푏 − 1 2(푏 − 2) Skewness 2푎[2푎2 + 3푎 푏 − 1 + (푏 − 1)2]

푏 − 1 3 푏 − 2 (푏 − 3) Kurtosis 3a(ab3 + 2b3+2a2b2 + 5ab2 − 6b2+a3b + 8a2b − 13ab + 6b + 5a3 − 10a2 +7a − 2)

b − 1 4 b − 2 b − 3 b − 4

229

A.10 Lomax (Pareto II) (a=1 in table A.9) pdf 푏 푓 푥 = , 푏 > 0, 0 < 푥 < ∞ 1 + x 1+b rth order moment Γ 1 + 푟 Γ 푏 − 푟

Γ 푏 Mean 1

푏 − 1

Mode 푏 − 1

2 Variance 푏

(푏 − 2)(푏 − 1)2 Skewness 2푏 푏 + 1

푏 − 1 3 푏 − 2 푏 − 3 Kurtosis 3(3b3 + b2 + 2b)

b − 1 4 b − 2 b − 3 b − 4

A.11 Log Logistic (Fisk) (a=b=1 in table A.9) pdf 1 푓 푥 = , 0 < 푥 < ∞ 1 + 푥 2

th rπ r order moment sin rπ

π Mean sin π

Mode 0 Variance 2π π 2 − sin 2π sin π

Skewness 3π 2π π π 3 − 3 + 2 sin 3π sin 2π sin π sin π

Kurtosis 4π π 3π π 2 2π − 4 + 6 sin 4π sin π sin 3π sin π sin 2π π 4 − 3 sin π

230

퐲퐜 A.12 Libby and Novick (1982) Beta Distribution (퐱 = in table A.1) ퟏ−퐲+퐲퐜 pdf 1 푐푎 푦푎−1 1 − 푦 푏−1 푓 푥 = , B a, b 1 − 1 − c y a+b 푎 > 0, 푏 > 0, 푐 > 0, 0 < 푥 < 1

퐲 A.13 McDonald’s (1995) beta distribution of the first kind (퐱 = in table A.1) 퐜

푏−1 Pdf 푎−1 푥 푥 1 − 푐 푓 푥 = , 푐푎 B a, b 푎 > 0, 푏 > 0, 푐 > 0, 0 < 푥 < 푐

퐜퐲 A.14 Three Parameter Beta Distribution (퐱 = in table A.1) ퟏ+퐲

Pdf 푐푎 푥푎−1 1 + (1 − 푐)푥 푏−1 푓 푥 = , B a, b (1 + 푥)a+b 푎 > 0, 푏 > 0, 푐 > 0, 0 < 푥 < 1

퐲 A.15 Chotikapanich et.al (2010) Beta of the Second Kind Distribution (퐱 = in 퐜 table A.9)

Pdf 푥푎−1 푓 푥 = , 푥 푎+푏 푎 푐 B a, b 1 + 푐 푎 > 0, 푏 > 0, 푐 > 0, 0 < 푥 < ∞

A.16 General Three Parameter Beta Distribution (퐱 = 퐜퐲 in table A.9)

Pdf 푐푎 푥푎−1 푓 푥 = , B a, b 1 + 푐푥 푎+푏 푎 > 0, 푏 > 0, 푐 > 0, 0 < 푥 < ∞

231

A.17 Three Parameter Beta Distribution (퐱 = 퐲퐜 in table A.1)

Pdf 푐푥푎푐−1 1 − 푥푐 푏−1 푓 푥 = , B a, b 푎 > 0, 푏 > 0, 푐 > 0, 0 < 푥 < 1

A.18 Generalized Four Parameter Beta Distribution of the First Kind

퐲 퐜 (퐱 = 퐢퐧 퐭퐚퐛퐥퐞 퐀. ퟏ) 퐪

Pdf 푥 푝 푏−1 푝푎−1 푝 푥 1 − 푞 푓 푥 = , 푎 > 0, 푏 > 0, 푝 > 0, 푞푎푝 퐵 푎, 푏 푞 > 0, 0 ≤ 푥푝 ≤ 푞푝

th 푟 r order moment 푟 푞 Γ a + b Γ 푎 + 푝 푟

Γ 푎 + 푏 + 푝 Γ a

1 Mean 푞Γ a + b Γ a + 푝

Γ a + b + 푝 Γ a

Mode 1 푞푝 푝푎 − 1 푝

푝푎 + 푝푏 − 1 − 푝

2 1 Variance 2 2 2 Γ a + p Γ b Γ a + b Γ a + p Γ b Γ a + b 푞2 − 2 2 1 Γ a + b + p Γ a + b + p Skewness 3 1 2

Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 푞3 − 3 3 1 2 Γ a + b + Γ a Γ a + b + Γ a Γ a + b + Γ a 푝 푝 푝 1 3

Γ a + b Γ a + 푝 + 2 1

Γ a + b + 푝 Γ a

232

Kurtosis 4 1 3

Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 q4 − 4 4 1 3 Γ a + b + Γ a Γ a + b + Γ a Γ a + b + Γ a 푝 푝 푝 1 2 2

Γ a + b Γ a + 푝 Γ a + b Γ a + 푝 + 6 1 2

Γ a + b + 푝 Γ a Γ a + b + 푝 Γ a 1 4 Γ a + b Γ a + 푝 − 3 1

Γ a + b + 푝 Γ a

A.19 Pareto Type I (b=1, p=-1 in table A.1) Pdf 푎푞푎 푓 푥 = , 푎 > 0, 푞 > 0, 푥푎+1 0 < 푞 ≤ 푥

Cumulative distribution function 푞 푎 1 − 푥 rth order moment 푞푟 a

푎 − 푟

푞푎 Mean 푎 − 1

Mode 푞

Variance 푞2a

a − 2 a − 1 2

Skewness 2(1 + a) a − 2

a − 3 a Kurtosis 3(3a3 − 5a2 − 4)

a a − 3 a − 4

233

A.20 Stoppa (Generalized Pareto Type I) (a=1, p=-p in table A.1) Pdf 푥 −푝 b−1 푓 푥 = p푞푝푏푥−푝−1 1 − , 푏 > 0, 푝 > 0, 푞 > 0, 푞 0 < 푞 ≤ 푥 Cumulative 푥 −푝 b 1 − distribution function 푞 th 푟 r order moment r q 푏Γ 푏 Γ 1 − 푝 푟 Γ 1 + 푏 − 푝 1 Mean 푞bΓ 푏 Γ 1 − 푝

1 Γ 1 + 푏 − 푝 Mode 1 1 + 푝푏 푝 푞 1 + 푝 2 1 Variance 2 2 2 Γ 1 − p Γ b Γ 1 + b Γ 1 − p Γ b Γ 1 + b q2 − 2 2 1 Γ 1 + b − p Γ 1 + b − p

Skewness 3 1 2

bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 푞3 − 3 3 3 1 1 2 2 b − Γ b − b − Γ b − b − Γ b − 푝 푝 푝 푝 푝 푝 1 3 bΓ b Γ 1 − 푝 + 2 1 1 b − 푝 Γ b − 푝

Kurtosis 4 1 3

bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 bΓ b Γ 1 − 푝 q4 − 4 4 1 3 Γ 1 + b − Γ 1 + b − Γ 1 Γ 1 + b − 푝 푝 푝 1 2 2

Γ 1 + b Γ 1 − 푝 Γ 1 + b Γ 1 − 푝 + 6 1 2 Γ 1 + b − 푎 Γ 1 + b − 푝 1 4

Γ 1 + b Γ 1 + 푝 − 3 1 Γ 1 + b + 푝

234

A.21 Kumaraswamy (a=1, q=1 in table A.1)

Pdf 푓 푥 = pb푥푝−1 1 − 푥푝 b−1, 푏 > 0, 푝 > 0

rth order moment 푟 bΓ 푝 + 1 Γ(푏) 푟 Γ(1 + 푝 + 푏)

Mean 1 bΓ 푝 + 1 Γ(푏)

1 Γ(1 + 푝 + 푏)

Mode 1 1 − 푝 푝

1 − 푝푏

Variance 2 2 1 p ! p ! b! b − 1 ! − b! b − 1 ! 2 1 + b ! + b ! p p

3 1 2 Skewness bΓ b Γ a + 푝 bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 − 3 3 1 2 Γ 1 + b + 푝 Γ 1 + b + 푝 Γ 1 + b + 푝 1 3

bΓ b Γ 1 + 푝 + 2 1 Γ 1 + b + 푝 4 1 3 Kurtosis bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 − 4 4 1 3 Γ 1 + b + 푝 Γ 1 + b + 푝 Γ 1 + b + 푝 1 2 2

bΓ b Γ 1 + 푝 bΓ b Γ 1 + 푝 + 6 1 2 Γ 1 + b + 푝 Γ 1 + b + 푝 1 4

bΓ b Γ 1 + 푝 − 3 1 Γ 1 + b + 푝

235

A.22 Generalized Power (b=1, p=1 in table A.1) pdf 푎푥푎−1 푓 푥 = , 푎 > 0, 푞 > 0, 0 < 푥 < 푞 푞푎 rth order moment 푞푟 a

a + r

푞푎 Mean a + 1

Variance (a + 1)2 a + 2 aΓ a − a2Γ2 a 푞2 (a + 1)2 a + 2 2

Skewness 3 2푎 −a3 − 3a2 + 4 1 2

a + 3 a + 1 2 a + 2 aΓ a − a2Γ2 a

Kurtosis (a + 1)2 a + 2 2 2 a 4a2 . q4 − 푞2(a + 1)2 a + 2 aΓ a − a2Γ2 a a + 4 (a + 1)(a + 3) 6a3 3a4 + − a + 1 2(a + 2) a + 1 4

236

A.23 Generalized Four Parameter Beta Distribution of the Second Kind

퐱 퐩 (퐱 = in table A.9) 퐪

Pdf 푝 푥푝푎−1 푓 푥 = 푥 , 푎 > 0, 푏 > 0, 푝 > 0, 푝푎 푝 a+b 푞 퐵 푎, 푏 (1 + (푞) ) 푞 > 0, 0 < 푥 < ∞ th 푟 푟 r order moment 푞푟 Γ 푎 + Γ 푏 − 푝 푝

Γ(푎)Γ(푏) Mean 1 1 푞Γ 푎 + 푝 Γ 푏 − 푝

Γ 푎 Γ 푏 Mode 1 푝푎 − 1 푝 푞 푝푏 + 1 Variance 2 2 2 1 1 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 푞2 − 퐵 푎, 푏 퐵 푎, 푏

Skewness 3 3 1 1 2 2 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 푞3 − 3 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏

1 1 3 퐵 푎 + 푝 , 푏 − 푝 + 2 퐵 푎, 푏

Kurtosis 4 4 1 1 3 3 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 푞4 − 4 퐵 푎, 푏 퐵 푎, 푏 퐵 푎, 푏

1 1 2 2 2 퐵 푎 + 푝 , 푏 − 푝 퐵 푎 + 푝 , 푏 − 푝 + 6 퐵 푎, 푏 퐵 푎, 푏

1 1 4 퐵 푎 + 푝 , 푏 − 푝 − 3 퐵 푎, 푏

237

A.24 Singh Maddala (Burr 12) (풂 = ퟏ in table A.23)

Pdf 푝 푏푥푝−1 푓 푥 = , 푏 > 0, 푝 > 0, 푞 > 0 푝 푥 푝 1+b 푞 1 + (푞) 0 < 푥 < ∞

Cumulative 푥 퐹 푥 = 1 − 1 + ( )푝 –b distribution function 푞

th r order moment 푟 푟 푟 푞 Γ 1 + 푝 Γ 푏 − 푝

Γ b

Mean 1 1 푞Γ 1 + 푝 Γ 푏 − 푝

Γ b

Mode 1 푝 − 1 푝 푞 푝푏 + 1

Variance 2 2 2 1 2 1 Γ(b)Γ 1 + 푝 Γ 푏 − 푝 − Γ (1 + 푝)Γ 푏 − 푝 푞2 Γ2(푏)

Skewness 3 3 1 1 2 2 푞3 푏퐵 1 + , 푏 − − 3푏퐵 1 + , 푏 − 푏퐵 1 + , 푏 − 푝 푝 푝 푝 푝 푝

3 1 1 + 2 푏퐵 1 + , 푏 − 푝 푝

Kurtosis 4 4 1 1 3 3 푞4 푏퐵 1 + , 푏 − − 4푏2퐵 1 + , 푏 − 퐵 1 + , 푏 − 푝 푝 푝 푝 푝 푝

2 1 1 2 2 + 6 푏퐵 1 + , 푏 − 푏퐵 1 + , 푏 − 푝 푝 푝 푝 4 1 1 − 3 푏퐵 푎 + , 푏 − 푝 푝

238

ퟏ A.25 Generalized Lomax (Pareto II) (a=1, p=1, q = in table A.23) 훌

Pdf 휆푏 푓 푥 = , 푏 > 0, 휆 > 0, 0 < 푥 < ∞ 1 + 휆푥 1+b

Cumulative 퐹 푥 = 1 − 1 + 휆푥 −b distribution function rth order moment 1 푟 Γ(1 + 푟)Γ(푏 − 푟) 휆 Γ(b)

Mean 1

휆 b − 1

Mode 0 Variance b

휆2 b − 1 2 b − 2

Skewness 2(b + 1) b − 2

(b − 3) b

Kurtosis 1 4 24 24 − 휆 b − 1 b − 2 b − 3 (b − 4) (b − 1)2 b − 2 (b − 3) 12 3 + − (b − 1)3(b − 2) (b − 1)4

239

A.26 Inverse Lomax (b=1, p=1 in table A.23) Pdf 푎푥푎−1 푓 푥 = 1+a , 푎 > 0, 푞 > 0, 0 < 푥 < ∞ 푎 푥 푞 1 + 푞 rth order moment 푞 푟 Γ(푎 + 푟)Γ(1 − 푟)

Γ(푎)Γ(1) Mode 푎 − 1 푞 2 Variance 푞2 푎퐵 푎 + 2, −1 − 푎퐵 푎 + 1,0 2 Skewness 퐵 푎 + 3,1 − 3 퐵 푎 + 1,1 − 1 퐵 푎 + 2,1 − 2 푞3 − 3 퐵 푎, 1 퐵 푎, 1 퐵 푎, 1 퐵 푎 + 1,1 − 1 3 + 2 퐵 푎, 1 Kurtosis 퐵 푎 + 4,1 − 4 퐵 푎 + 1,1 − 1 퐵 푎 + 3,1 − 3 푞4 − 4 퐵 푎, 1 퐵 푎, 1 퐵 푎, 1 퐵 푎 + 1,1 − 1 2 퐵 푎 + 2,1 − 2 + 6 퐵 푎, 1 퐵 푎, 1 퐵 푎 + 1,1 − 1 4 − 3 퐵 푎, 1

A.27 Dagum (Burr 3) (b=1 in table A.23) Pdf 푝 푎푥푝푎−1 푓 푥 = , 푎 > 0, 푝 > 0, 푞 > 0 푝푎 푥 푝 1+a 푞 (1 + (푞) ) 0 < 푥 < ∞ th r order moment 푟 푟 푟 푞 Γ(푎 + 푝)Γ(1 − 푝)

Γ(a) Mean 1 1 푞Γ(푎 + 푝)Γ(1 − 푝)

Γ(a) Mode 1 푝푎 − 1 푝 푞 푝 + 1 Variance 2 2 2 1 2 1 Γ(a)Γ 푎 + 푝 Γ 1 − 푝 − Γ (푎 + 푝)Γ 1 − 푝 푞2 Γ2(푎)

240

A.28 Para-Logistic (a=1, b = p in table A.23) Pdf 푝2푥푝−1 푓 푥 = , 푝 > 0, 푞 > 0 푝 푥 푝 1+p 푞 (1 + (푞) ) 0 < 푥 < ∞

ퟏ A.29 General Fisk (Log –Logistic) (a=b=1, q = in table A.23) 흀 pdf 휆 푝 휆푥 푝−1 푓 푥 = , 푝 > 0, 휆 > 0, 0 < 푥 < ∞ (1 + (휆푥)푝 )2 rth order moment 푟 푟 푞푟 Γ 1 + Γ 1 − 푝 푝 Mean 1 1 1 푞 Γ Γ 1 − 푝 푝 푝 Mode 1 푝 − 1 푝 푞 푝 + 1 Variance 2 2 1 1 푞2 Γ 1 + Γ 1 − − Γ2(1 + )Γ2 1 − 푝 푝 푝 푝

풖 풗 풗 A.30 Fisher (F) (a= , b= , q = in table A.23) ퟐ ퟐ 풖

푢 푢 pdf 푢 + 푣 푢 2 −1 2 Γ 2 푣 푥 푓 푥 = 푢+푣 , 푢 > 0, 푣 > 0 푢 푣 푥푢 2 Γ 2 Γ 2 1 + 푣 0 < 푥 < ∞ rth order moment 푣 푟 푢 푣 푢 Γ 2 + 푟 Γ 2 − 푟 푢 푣 Γ 2 Γ 2 푣 Mean 푣 − 2 Mode 푢 − 2 푣

푢 푣 + 2

Variance 푢 + 푣 − 2 2푣2 푢 푣 − 2 2 푣 − 4

241

ퟏ 풓 A.31 Half Student’s t (a= , b= , q = 풓 in table A.23) ퟐ ퟐ

1 + 푟 pdf 2Γ 2 푓 푥 = 1+푟 , −∞ < 푥 < ∞ 푟 푥2 2 rπΓ 2 1 + 푟

A.32 Logistic (x = 풆풚in table A.23) pdf 푝 푒푝푎(푥−푙표푔 ⁡(푞) 푓 푥 = , −∞ < 푥 < ∞ 퐵 푎, 푏 (1 + 푒푝(푥−log 푞 ))푎+푏

ퟏ A.33 Generalized Gamma (퐪 = 휷풃풑, 퐥퐢퐦 in table A.23) 풃→∞

푝푎−1 푥 푝 pdf 푝 푥 − 푓 푥 = 푒 훽 , 푎 > 0, 푝 > 0, β > 0, β푝푎 Γ 푎 0 < 푥 < ∞ th 푟 r order moment β푟 Γ 푎 + 푝

Γ 푎 1 Mean βΓ 푎 + 푝

Γ 푎 Mode 1 1 푝 훽 푎 − 푝 Variance 훽2 2 1 Γ a Γ 푎 + − Γ2 푎 + Γ2 a 푝 푝

Skewness 3 2 3 3 2 1 3 3 1 β Γ Γ β Γ Γ β Γ 푎 푎 + 푝 − 3 Γ a 푎 + 푝 푎 + 푝 + 2 푎 + 푝

Γ3 푎

Kurtosis 4 3 1 2 1 Γ Γ Γ Γ Γ2 푎 + 푝 4 푎 + 푝 푎 + 푝 6 푎 + 푝 푎 + 푝 β4 − + Γ 푎 Γ2 푎 Γ3 푎 1 Γ4 푎 + 푝 − 3 Γ4 푎

242

A.34 Gamma (퐩 = ퟏ, in table A.33)

푥 pdf 푥 푎 −1 − 푓 푥 = 푒 훽 , 푎 > 0, β > 0, 0 < 푥 < ∞ β푎 Γ 푎 rth order moment β푟 Γ 푎 + 푟

Γ 푎

Mean Β푎

Mode β a − 1

Variance a훽2

Skewness 2

푎 Kurtosis 6 3 + 푎

A.35 Inverse Gamma (퐩 = −ퟏ, in table A.33)

푎 훽 pdf β − 푓 푥 = 푒 푥 , 푎 > 0, β > 0, 0 < 푥 < ∞ 푥1+푎 Γ 푎 rth order moment β푟 Γ 푎 − 푟

Γ 푎

Mean β

a − 1

Mode β

a + 1

Variance 훽2

푎 − 1 2 a − 2

Skewness 4 (a − 2)

a − 3

Kurtosis 3(푎 − 2)(푎 + 5)

푎 − 3 푎 − 4

243

A.36 Weibull (퐚 = ퟏ, in table A.33)

푥 푝 pdf 푝 − 푓 푥 = 푥푝−1 푒 훽 , 푝 > 0, β > 0, 0 < 푥 < ∞ β푝

푦 푝 Cumulative − distribution function 1 − 푒 훽 th r order moment 푟 푟 β Γ 1 + 푝

Mean β 1 Γ p p Mode 1 푝 − 1 푝 훽 푝 Variance 2 β2Γ 1 + − μ2 푝 A.37 Exponential (퐚 = 퐩 = ퟏ, in table A.33)

푥 pdf 1 − 푓 푥 = 푒 훽 , β > 0, 0 < 푥 < ∞ β 푥 Cumulative −( ) 훽 distribution function 퐹 푥 = 1 − 푒 rth order moment β푟 rΓ 푟 Mean β Mode 0 Variance β2

퐧 A.38 Chi-Squared (퐚 = , 퐩 = ퟏ, 훃 = ퟐ, in table A.33) ퟐ

n 푥 pdf 1 −1 − 푓 푥 = 푥2 푒 2, 0 < 푥 < ∞ n n 2Γ 2 2 th n r order moment 푟 Γ 2 2 + 푟 n Γ 2

Mean n Mode n − 2 Variance 2n

244

A.39 Rayleigh (퐚 = ퟏ, 퐩 = ퟐ, 훃 = 훂 ퟐ, in table A.33)

푥 2 pdf 푥 − 푓 푥 = 푒 α 2 , α > 0, 0 < 푥 < ∞ α2 Cumulative 푥 2 − 2 distribution function 퐹(푥) = 1 − 푒 2α th 푟 푟 r order moment α 2 Γ 1 + 2 Mean π α 2 Mode α Variance 4 − 휋 α2 2

ퟑ A.40 Maxwell (퐚 = , 퐩 = ퟐ, 훃 = 훂 ퟐ, in table A.33) ퟐ

푥 2 pdf − 2 푥2푒 2∝2 푓 푥 = , α > 0, 0 < 푥 < ∞ π α3 rth order moment 푟 3 푟 α 2 Γ + 2 2 1 2 π Mean 2α 2

π Mode 2α

Variance 3π − 8 α2 π

ퟏ A.41 Half Standard Normal (퐚 = , 퐩 = ퟐ, 훃 = ퟐ, in table A.33) ퟐ

푥 2 pdf 2 − 푓 푥 = 푒 2 , 0 < 푥 < ∞ 2π rth order moment 푟 1 푟 2 Γ + 2 2 1 Γ 2 Mean 2

π Mode 0 Variance 1

245

ퟏ A.42 Half Normal (퐚 = , 퐩 = ퟐ, 훃 = 훔 ퟐ, in table A.33) ퟐ

푥 2 pdf − 2 푒 2휍2 푓 푥 = , 0 < 푥 < ∞ π 휍 rth order moment 푟 1 푟 ς 2 Γ + 2 2 1 Γ 2 Mean ς 2

π Mode 0

Variance 2 ς2 1 − π

ퟏ A.43 Half Log Normal (풙 = (풍풏 풚 − 흁, 풂 = , 훃 = 흈 ퟐ in table A.33) ퟐ

ln 푥 −휇 2 pdf 2 − 푓 푥 = 푒 2ς2 , 0 < 푥 < ∞ ς푥 2π th 1 r order moment 푟휇 + 푟 2휍2 e 2 1 Mean 휇 + 휍2 e 2 2 Mode e휇 +휍

2 2 Variance e2휇 +휍 e휍 − 1

풚−풑 A.44 Four Parameter Generalized Beta Distribution (풙 = in table A.1) 풒−풑 pdf 푥 − 푝 푎−1 푞 − 푥 푏−1 푓 푥 = , 푎, 푏, 푝, 푞 > 0, 0 < 푥 < ∞ 퐵 푎, 푏 푞 − 푝 푎+푏−1

Mean 푎푞 + 푏푝

푎 + 푏 Mode 푎 − 1 푞 + (푏 − 1)푝

푎 + 푏 − 2

Variance 푎푏(푞 − 푝)2

푎 + 푏 2(푎 + 푏 + 1)

246

풚풑 A.45 Five Parameter Generalized Beta Distribution (풙 = in table A.1) 풒풑+풄풚풑 pdf 푥 푝 푏−1 푎푝−1 푝푥 1 − 1 − 푐 푞 푓 푥 = , 푎, 푏, 푝, 푞 > 0, 0 < 푐 < 1 푥 푝 푎+푏 푎푝 푞 퐵 푎, 푏 1 + 푐 푞

th 푟 푟 r order moment 푟 푟 푎 + , ; 푐 푞 퐵(푎 + 푝 , 푏) 푝 푝 2퐹1 푟 퐵(푎, 푏) 푎 + 푏 + 푝

Mean 1 1 1 푎 + , ; 푐 푞퐵(푎 + 푝 , 푏) 푝 푝 퐹 퐵(푎, 푏) 2 1 1 푎 + 푏 + 푝

Mode 푎 − 1 푞 + (푏 − 1)푝

푎 + 푏 − 2

Variance 2 2 2 q2B(a + , b) a + , ; c p p p 2 2F1 a+ 2 1 − c pB(a, b) a + b + p 2 1 1 1 qB(a + , b) a + , ; c p p p − 1 2F1 a+ 1 1 − c p B(a, b) a + b + p

247

풌 A.46 Five Parameter Weighted Generalized Beta Distribution (풂 = 풂 + , 풃 = 풑 풌 풃 − in table A.18) 풒 pdf 푝푥푎푝+푘−1 푓 푥 = , 푎, 푏, 푝, 푞 > 0, 푝 푎+푏 푎푝 +푘 푘 푘 푥 푞 퐵 푎 + 푝 , 푏 − 푝 1 + 푞 0 < 푥 < ∞, −푎푝 < 푘 < 푏푝 th r order moment 푟 푘 푟 푘 푟 푞 퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝

푘 푘 퐵 푎 + 푝 , 푏 − 푝 Mean 푘 1 푘 1 푞퐵 푎 + 푝 + 푝 , 푏 − 푝 − 푝

푘 푘 퐵 푎 + 푝 , 푏 − 푝 Variance 2 푘 + 2 푘 + 2 푘 + 1 푘 − 1 퐵 푎 + 푝 , 푏 − 푝 푎 + 푝 , 푏 − 푝 푞푟 − 푘 푘 푘 푘 퐵 푎 + , 푏 − 푎 + , 푏 − 푝 푝 푝 푝

A.47 Weighted Beta of the Second Kind (풑 = 풒 = ퟏ in table A.46) pdf 푥푎+푘−1 푓 푥 = , 푎, 푏, > 0, 푞푎+푘 퐵 푎 + 푘, 푏 − 푘 1 + 푥 푎+푏 0 < 푥 < ∞, −푎 < 푘 < 푏

A.48 Weighted Singh-Maddala (Burr12) (풂 = ퟏ in table A.46)

Pdf 푝푥푝+푘−1 푓 푥 = , 푏, 푝, 푞 > 0, 푝 1+푏 푝+푘 푘 푘 푥 푞 퐵 1 + 푝 , 푏 − 푝 1 + 푞 0 < 푥 < ∞, −푝 < 푘 < 푏푝

A.49 Weighted Dagum (Burr3) (풃 = ퟏ in table A.46)

Pdf 푝푥푎푝+푘−1 푓 푥 = , 푎, 푝, 푞 > 0, 푝 푎+1 푎푝+푘 푘 푘 푥 푞 퐵 푎 + 푝 , 1 − 푝 1 + 푞 0 < 푥 < ∞, −푎푝 < 푘 < 푝

248

ퟏ A.50 Weighted Generalized Fisk (Log-Logistic) (풂 = 풃 = ퟏ, 풒 = in table 흀 A.46)

Pdf 푝푥푝+푘−1 푓 푥 = , 푝, 푞, 휆 > 0, 1 푝+푘 푘 푘 퐵 1 + , 1 − 1 + (휆푥)푝 2 휆 푝 푝 0 < 푥 < ∞, −푝 < 푘 < 푝

A.51 Five Parameter Exponential Generalized Beta Distribution (풙 = 풆풚, 풑 = ퟏ and 풒 = 풆휹 in table A.18) 흈

Pdf 푎 푥−훿 푥−훿 푏−1 푒 휍 1 − 1 − 푐 푒 휍 푓 푥 = , 푎, 푏, 푐, 휍, 훿 > 0, 1 푎+푏 푒푥 휍 휍퐵 푎, 푏 1 + 푐 푒훿 푥 − 훿 1 −∞ < < ln 휍 1 − 푐

A.52 Exponential Generalized Beta Distribution of the First Kind (풄 = ퟎ in table A.51)

Pdf 푎 푥−훿 푥−훿 푏−1 푒 휍 1 − 푒 휍 푓 푥 = , 푎, 푏, 휍, 훿 > 0, 휍퐵 푎, 푏 푥 − 훿 −∞ < < 0 휍

A.53 Exponential Generalized Beta Distribution of the Second Kind (풄 = ퟏ in table A.51)

Pdf 푎 푥−훿 푒 휍 푓 푥 = , 푎, 푏, 휍, 훿 > 0, 푥−훿 푎+푏 휍퐵 푎, 푏 1 + 푒 휍 푥 − 훿 −∞ < < ∞ 휍

249

A.54 Exponential Generalized Gamma (Generalized Gompertz) (풃 → ∞ 퐚퐧퐝 휹∗ = 흈풍풏풒 + 휹 in table A.51)

Pdf 푎 푥−훿 푒 휍 푒−푒 푥−훿 /휍 푓 푥 = , 푎, 휍, 훿 > 0, 휍ΓΞ푝 푥 − 훿 −∞ < < ∞ 휍

A.55 Generalized Gumbell (풄 = ퟏ 퐚퐧퐝 풂 = 풃 in table A.51)

Pdf 푒푎 푥−훿 /휍 푓 푥 = , 푎, 휍, 훿 > 0, 푥−훿 2푎 휍퐵 푎, 푎 1 + 푒 휍

푥 − 훿 −∞ < < ∞ 휍

A.56 Exponential Power (풄 = ퟎ 퐚퐧퐝 풃 = ퟏ in table A.51)

Pdf 푎푒푎 푥−훿 /휍 푓 푥 = , 푎, 휍, 훿 > 0, 휍

푥 − 훿 −∞ < < 0 휍

A.57 Exponential (Two Parameter) (풄 = ퟎ, 풂 = ퟏ 퐚퐧퐝 풃 = ퟏ in table A.51)

Pdf 푒 푥−훿 /휍 푓 푥 = , 휍, 훿 > 0, 휍

푥 − 훿 −∞ < < 0 휍

250

A.58 Exponential (One Parameter) (풄 = ퟎ, 풂 = ퟏ, 풃 = ퟏ 퐚퐧퐝 휹 = ퟎ in table

A.51)

Pdf 푒푥/휍 푓 푥 = , 휍 > 0, 휍

푥 −∞ < < 0 휍

A.59 Exponential Fisk (Logistic) (풄 = ퟏ, 풂 = ퟏ, 풃 = ퟏ in table A.51)

Pdf 푒(푥−훿)/휍 푓 푥 = , 휍 > 0, 휍(1 + 푒(푥−훿)/휍 )2

푥 − 훿 −∞ < < ∞ 휍

A.60 Standard Logistic (풄 = ퟏ, 풂 = ퟏ, 풃 = ퟏ, 휹 = ퟎ in table A.51)

Pdf 푒푥/휍 푓 푥 = , 휍 > 0, 휍(1 + 푒푥/휍 )2

푥 −∞ < < ∞ 휍

A.61 Exponential Weibull (풃 → ∞, 휹∗ = 흈풍풏풒 + 휹 퐚퐧퐝 풂 = ퟏ in table A.51)

Pdf 푥−훿 푒 휍 푒−푒 푥−훿 /휍 푓 푥 = , 휍, 훿 > 0, 휍ΓΞ푝 푥 − 훿 −∞ < < ∞ 휍

Other Generalized Beta Distributions

A.62 Beta Generator (풙 = 푮 풚 in table A.1)

푎−1 푏−1 Pdf 푔(푥) 퐺 푥 1 − 퐺 푥 푓 푥 = , 푎 > 0, 푏 > 0, 퐵(푎, 푏) −∞ < 푥 < ∞, 0 < 퐺(푥) < 1

251

A.63 ith Order Statistics (a=i, b=n-i+1 in table A.62) Pdf n! 푓 푥 = 푔(푥)[퐺(푥)]푖−1 1 − 퐺 푥 푛−푖 , 푖 > 0, 푖 − 1 ! 푛 − 푖 ! 푛 − 푖 > 0, −∞ < 푥 < ∞, 0 < 퐺(푥) < 1

A.64 Beta-Normal (Substituting G(X) and g(X) in table A.62)

Pdf 푥 − 휇 푎−1 푥 − 휇 푏−1 휍−1 Φ 1 − Φ 푥 − 휇 푓 푥 = 휍 휍 휙 퐵 푎, 푏 휍 푎 > 0, 푏 > 0, 휍 > 0, 휇 ∈ ℝ 푎푛푑 푥 ∈ ℝ

x − μ Parent cdf 퐺 푥 = Φ ς

x − μ Parent pdf 푔 푥 = 휙 ς

A.65 Beta-Exponential (Substituting G(X) and g(X) in table A.62) Pdf λ exp(−푏λx)[1 − exp(−λx)]푎−1 푓 푥 = 퐵 푎, 푏 푎 > 0, 푏 > 0, λ > 0, 푥 >0

Parent cdf 퐺 푥 = 1 − exp(−λx) Parent pdf 푔 푥 = λexp(−λx)

A.66 Beta-Weibull (Substituting G(X) and g(X) in table A.62) 푎−1 Pdf 푥 푐 x c 푐 −푏 − 푥푐−1 푒 훽 1 − e β β푐 푓 푥 = , 푎, 푏, 푐, β > 0 퐵 푎, 푏

x c Parent cdf − 퐺 푥 = 1 − e β x c Parent pdf 푐 − 푔 푥 = 푥푐−1e β β푐

252

A.67 Beta-Hyperbolic Secant (Substituting G(X) and g(X) in table A.62)

푏−1 Pdf 2 푎−1 2 arctan 푒푥 1 − arctan 푒푥 푓 푥 = π π 퐵 푎, 푏 π cosh x 푎 > 0, 푏 > 0 and 푥 ∈ ℝ

Parent cdf 2 퐺 푥 = arctan 푒푥 π Parent pdf 1 푔 푥 = π cosh x

A.68 Beta-Gamma (Substituting G(X) and g(X) in table A.62) Pdf 푏−1 푥 Γx ρ − 푥휌−1푒 휆 Γx ρ 푎−1 1 − λ λ Γ ρ 푓 푥 = 퐵 푎, 푏 Γ ρ aλρ 푎, 푏, ρ, λ, x > 0

Parent cdf Γx ρ 퐺 푥 = λ Γ ρ 휌−1 푥 Parent pdf 푥 − 푔 푥 = 푒 휆 휆

A.69 Beta-Gumbel (Substituting G(X) and g(X) in table A.62) x − μ Pdf exp − x−μ x−μ b−1 ς −푎 (exp − ) − exp − 푓 푥 = e ς 1 − e ς 휍퐵 푎, 푏 푎, 푏, 푢, ς, 푥 > 0

x − μ Parent cdf 퐺 푥 = exp − exp − ς Parent pdf x − μ exp − x − μ 푔 푥 = ς exp −(exp − ) ς ς

253

A.70 Beta-Frèchet (Substituting G(X) and g(X) in table A.62)

b−1 Pdf x −λ x −λ λ λ휍 exp −a ς 1 − exp − ς 푓 푥 = 푥1+λ 퐵 푎, 푏 푎, 푏, λ, ς, x > 0

Parent cdf x −λ 퐺 푥 = exp − ς

Parent pdf λ휍λ x −λ 푔 푥 = exp − 푥1+λ ς

A.71 Beta-Maxwell (Substituting G(X) and g(X) in table A.62)

푥 2 Pdf a−1 b−1 − 1 2 3 푥2 2 3 푥2 2 푥2푒 2∝2 푓 푥 = γ , 1 − γ , 퐵 푎, 푏 π 2 2α π 2 2α π α3 푎, 푏, α, 푥 > 0 푎푛푑 γ 푎, 푏 is an incomplete gamma function

Parent cdf 2 3 푥2 퐺 푥 = γ , π 2 2α 푥 2 Parent pdf − 2 푥2푒 2∝2 푔 푥 = π α3

A.72 Beta-Pareto (Substituting G(X) and g(X) in table A.62)

Pdf 푘 푥 −k a−1 푥 −푘푏 −1 푓 푥 = 1 − 휃퐵 푎, 푏 휃 휃 푎, 푏, 푘, 휃, 푥 > 0

Parent cdf 푥 −k 퐺 푥 = 1 − 휃 Parent pdf 푘휃푘 푔 푥 = 푥푘+1

254

A.73 Beta-Rayleigh (Substituting G(X) and g(X) in table A.62)

a−1 2 Pdf 푥 2 푥 푥 − −푏 푓 푥 = 1 − 푒 2α2 푒 α 2 α2퐵 푎, 푏 푎, 푏, 푘, 훼 > 0

푥 2 Parent cdf − 퐺 푥 = 1 − 푒 2α2 푥 2 Parent pdf 푥 − 푔 푥 = 푒 α 2 α2

A.74 Beta-Generalized-Logistic of Type IV (Substituting G(X) and g(X) in table A.62) Pdf 1−a−b qx B a, b e a−1 b−1 −x 푓 푥 = −x p+q [B 1 (p, q)] [B e (q, p)] B a, b 1 − e 1+e−x 1+e−x 푎, 푏, p, q > 0, 푥 > 푅

Parent cdf B 1 (p, q) −x 퐺 푥 = 1+e B p, q Parent pdf eqx 푔 푥 = 1 + e−x p+q

A.75 Beta-Generalized-Logistic of Type I (Substituting q=1 in table A.74) Pdf pe−x[ 1 + e−ax p − 1]b−1 푓 푥 = B a, b 1 + e−x a+pb 푎, 푏, p > 0, 푥 > 푅

A.76 Beta-Generalized-Logistic of Type II (Substituting q=1 in table A.74) Pdf qe−bqx e−qx p 푓 푥 = 1 − B a, b 1 + e−x qb +1 B a, b 1 + e−x q 푎, 푏, q > 0, 푥 > 푅

A.77 Beta-Generalized-Logistic of Type III (Substituting q=1 in table A.74) Pdf 1−a−b −qx B q, q e a−1 b−1 −x 푓 푥 = −x 2q [B 1 (q, q)] [B e (q, q)] B a, b 1 + e 1+e−x 1+e−x 푎, 푏, q > 0, 푥 > 푅

255

A.78 Beta-Beta Prime (퐱 = 퐞−퐲 퐢퐧 퐭퐚퐛퐥퐞 퐀. ퟕퟒ)

Pdf 퐵 푝, 푞 1−푎−푏 푥푞−1 푥 푎−1 푏−1 푓 푥 = 푝+푞 [퐵 (푞, 푝)] [퐵 1 (푝, 푞)] 퐵 푎, 푏 1 + 푥 1+푥 1+푥 푎, 푏, 푞, > 0, 푥 > 푅

A.79 Beta-F (Substituting G(X) and g(X) in table A.62) Pdf 푓 푥 v a−1 v b−1 v 2 −1 −1 2 B a, b u x v u v u v = (u+v)/2 I x , I 1 , u v v u 2 2 v 2 2 B , 1 + x v 1+ x 2 2 1+ x u u u 푎, 푏, u, v, x > 0

Parent cdf 1 푣 푢 v 퐺 푥 = B x , 푢 푣 u 2 2 B , v 2 2 1+ x u q Parent pdf v v 2 −1 x2 u 푔 푥 = 푢 푣 (u+v)/2 B , v 2 2 1 + u x A.80 Beta-Burr XII (Substituting G(X) and g(X) in table A.62)

a−1 Pdf 푝푘푞−푝 푥푝−1 푥 p −푘 푥 p −푘푏 −1 푓 푥 = 1 − 1 + 1 + 퐵 푎, 푏 q q 푎, 푏, 푝, 푞, 푘, 푥 > 0.

Parent cdf 푥 p −푘 퐺 푥 = 1 − 1 + q Parent pdf 푥 p −푘−1 푔 푥 = 푝푘푞−푝 1 + 푥푝−1 q

A.81 Beta-Dagum (Substituting G(X) and g(X) in table A.62) Pdf 1 푝 푞푎푥−푝−1 푓 푥 = 1 + 푞푥−푝 −푎 a−1 1 − 1 + 푞푥−푝 −푎 b−1 퐵 푎, 푏 (1 + 푞푥−푝 )1+a 푎, 푏, 푝, 푞, 푥 > 0

Parent cdf 퐺 푥 = 1 + 푞푥−푝 −푎 Parent pdf 푝 푞푎푥−푝−1 푔 푥 = (1 + 푞푥−푝 )1+a

256

A.82 Beta-Fisk (Log-Logistic) (Substituting G(X) and g(X) in table A.62) Pdf 휆푝 휆푥 ap −1 푓 푥 = 퐵 푎, 푏 1 + 휆푥 p a+b 푎, 푏, 푝, 휆, 푥 > 0

Parent cdf 휆푥 p 퐺 푥 = 1 + 휆푥 p Parent pdf 휆푝 휆푥 p−1 푔 푥 = 1 + 휆푥 p 2

A.83 Beta-Generalized Half Normal (Substituting G(X) and g(X) in table A.62) α a−1 α b−1 α 1 푥 ퟐ휶 Pdf 푥 푥 2 α 푥 − 2 휍 2휙 휍 − 1 2 − 2휙 휍 π 푥 휍 푒 푓 푥 = 퐵 푎, 푏 푎, 푏, α, 휍 > 0

Parent cdf 푥 α 퐺 푥 = 2휙 − 1 휍 Parent pdf α 1 푥 ퟐ휶 2 α 푥 − 푔 푥 = 푒 2 휍 π 푥 휍

A.84 Generalized Beta Generator (퐱 = 퐆 퐲 훂 in table A.1)

Pdf 푔 푥 훼 퐺 푥 훼−1 퐺 푥 훼 푎−1 1 − 퐺 푥 훼 푏−1 푓 푥 = , 푎 > 0, 퐵(푎, 푏) 훼 > 0, 푏 > 0, −∞ < 푥 < ∞, 0 < 퐺 푥 훼 < 1

A.85 Exponentiated Exponential (Substituting G(X) and g(X) in table A.84)

푎−1 푏−1 Pdf 푥 훼 푥 훼 푥 훼 − − − 1 − 푒 훽 1 − 푒 훽 1 − 1 − 푒 훽 푓 푥 = 퐵 푎, 푏

푥 훼 Parent cdf − 퐺 푥 = 1 − 푒 훽

푥 훼−1 푥 Parent pdf 훼 − − 푔 푥 = 1 − 푒 훽 푒 훽 훽

257

A.86 Exponentiated Weibull (Substituting G(X) and g(X) in table A.84)

푏−1 Pdf 푦 푝 푦 푝 푦 푝 푝 − − − 푦푝−1푒 훽 [1 − 푒 훽 ]푎−1 1 − 1 − 푒 훽 βp 푓 푥 = 퐵 푎, 푏

푦 푝 Parent cdf − 퐺 푥 = 1 − 푒 훽 푦 푝 Parent pdf 푝 − 푔 푥 = 푦푝−1푒 훽 βp

A.87 Exponentiated Pareto (Substituting G(X) and g(X) in table A.84)

푏−1 Pdf 푦 −푝 푏−1 푦 −푝 푏 푦 −푝 푏 푝qp 푏푦−푝−1 1 − [ 1 − ]푎−1 1 − 1 − 푞 푞 푞 푓 푥 = 퐵 푎, 푏

Parent cdf 푦 −푝 푏 퐺 푥 = 1 − 푞 Parent pdf 푦 −푝 푏−1 푔 푥 = 푝qp 푏푦−푝−1 1 − 푞

A.88 Generalized Beta Exponential (Substituting G(X) and g(X) in table A.84) Pdf 푝 λ exp(−λx)[1 − exp(−λx) ]푝푎−1 1 − 1 − exp −λx 푝 푏−1 푓 푥 = 퐵 푎, 푏

Parent cdf 퐺 푥 = 1 − exp(−λx) Parent pdf 푔 푥 = λ exp(−λx)

A.89 Generalized Beta Weibull (Substituting G(X) and g(X) in table A.84)

푝 푏−1 Pdf 푥 푐 x c x c 푐 − − − 푝 푥푐−1 푒 훽 [1 − e β ]푝푎 −1 1 − 1 − e β β푐 푓 푥 = 퐵 푎, 푏

x c Parent cdf − 퐺 푥 = 1 − e β 푥 푐 Parent pdf 푐 − 푔 푥 = 푥푐−1 푒 훽 β푐

258

A.90 Generalized Beta Hyperbolic Secant (Substituting G(X) and g(X) in table A.84) Pdf 2 푥−휇 푎푝−1 2 푥−휇 푝 푏−1 푝 arctan 푒 휍 1 − arctan 푒 휍 π π 푓 푥 = 푥 − 휇

퐵 푎, 푏 ςπ cosh 휍

Parent cdf 2 푥−휇 퐺 푥 = arctan 푒 휍 π Parent pdf 1 푔 푥 = 푥 − 휇 ςπ cosh 휍

A.91 Generalized Beta Normal (Substituting G(X) and g(X) in table A.84) Pdf 푥 − 휇 푥 − 휇 p 푏−1 푝휍−1[Φ ]푎푝−1 1 − Φ 푥 − 휇 푓 푥 = 휍 휍 휙 퐵 푎, 푏 휍

푥 − 휇 Parent cdf 퐺 푥 = Φ 휍 Parent pdf 푥 − 휇 푔 푥 = 휙 휍

A.92 Generalized Beta Log Normal (Substituting G(X) and g(X) in table A.84) Pdf 푝휙 log 푥 [Φ log 푥 ]푎푝−1 1 − Φ log 푥 푝 푏−1 푓 푥 = 푥퐵 푎, 푏

Parent cdf 퐺 푥 = Φ log 푥 Parent pdf 휙 log푥 푔 푥 = 푥

A.93 Generalized Beta Gamma (Substituting G(X) and g(X) in table A.84) Pdf 푎푝−1 푝 푏−1 ρ−1 x Γx ρ Γx ρ x − e λ λ 1 − λ λ Γ ρ Γ ρ 푓 푥 = 퐵 푎, 푏

Parent cdf Γx ρ 퐺 푥 = λ Γ ρ ρ−1 x Parent pdf x − 푔 푥 = e λ λ

259

A.94 Generalized Beta Frèchet (Substituting G(X) and g(X) in table A.84)

푎푝 −1 푝 푏−1 Pdf 푝λ x −λ x −λ x −λ exp − 푥휆−1 exp − 1 − exp − ς ς ς ς 푓 푥 = 퐵 푎, 푏

Parent cdf x −λ 퐺 푥 = exp − ς Parent pdf λ x −λ 푔 푥 = exp − 푥휆−1 ς ς

A.95 Confluent Hypergeometric Pdf 1 1 푓 푥 = 푢푎−1 1 − 푢 푐−푎−1푒푥푢푑푢 퐵(푎, 푐 − 푎) 0

A.96 Gauss Hypergeometric Pdf Γ(c) 1 푓 푥 = 푢푎−1(1 − 푢)푐−푎−1(1 − 푥푢)−푏푑푢 Γ(a)Γ(c − a) 0

A.97 Appell Pdf 퐶푥훼−1 1 − 푥 훽−1 푓 푥 = , 1 − 푢푥 휌 1 − 푣푥 휆 0 < 푥 < 1, 훼 > 0, 훽 > 0, 휌 > 0, 휆 > 0, −1 < 푢 < 1, −1 < 푣 < 1 and C is the normalizing constant given by 1 = 퐵 훼, 훽 퐹 (훼, 휌, 휆, 훼 + 훽; 푢, 푣) 퐶 1

260

SUMMARY: FRAMEWORK FOR CLASSICAL TO GENERALIZED BETA DISTRIBUTIONS

Power

Uniform Lomax (Pareto Type II) Beta of the first Kind Arcsine Beta of the Second Kind Log-logistic

(Fisk) Triangular Shaped

Parabolic

Wigner SemiCircle

3 Parameter generalized beta 3 Parameter generalized beta Libby&Novick‘s, McDonald‘s, Chotikapanich, McDonald‘s Beta type 3

4 Parameter generalized beta 4 Parameter generalized beta of the first kind of the second kind 5 Parameter generalized beta

(Pareto Type II) (Pareto Type GeneralizedLomax Lomax Inverse Dagum Para FiskGeneral (F) Fisher Logistic

Singh

Stoppa Kumaraswamy GeneralizedPower (TypeI) Pareto Gamma Unit

Logistic

Maddala

(Log

logistic)

Generalized Gamma

Half Log Half

Gamma Exponential Chi Rayleigh Maxwell Half Gamma Inverse Weibull

Squared -

Normal

261

11 References

1. Akinsete, Alfred, Felix Famoye, and Carl Lee. ―The beta Pareto distribution.‖ Statistics 42.6 (2008):547-563. 2. Akinsete,A., and C. Lowe. ―Beta Rayleigh distribution in reliability measure.Section on Physical and Engineering Sciences.‖ Proceedings of the American Statistical Association. (2009). 3. Alexander C., Cordeiro, G.M., Ortega, E.M.M., and Josè M.S (2010) ―Generalized Beta Generated Distributions‖ 4. Amusan, Grace Ebunoluwa, ―The Beta Maxwell Distribution (2010)”. 5. Barreto-Souza, W., Cordeiro, G. M., and Simas, A. B. Some results for beta Frèchet distribution, (2008). 6. C. Armero and M.J. Bayarri, Prior assessments for prediction in queues, The Statistician 43, 139-153, (1994). 7. Catherine, F., Merran, E., Nicholas, H., and Brian, P., Statistical distributions, 4th ed., Wiley, (2011) 8. Chen, J.C. and Novick, M.R., (1984), Bayesian analysis for binomial models with generalized beta prior distributions. Journal of Educational Statistics, 9, 163-175. 9. Chotikapanich, D., W.E. Griffiths, D.S.P Rao, and V. Valecia (2010). Global Income Distribution and Inequality, 1993 and 2000: Incorporating Country- Level Inequality Modeled with Beta Distributions. Forthcoming in The Review of economics and Statistics 10. Christian, C., and Samuel, K. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New Jersey: John Wiley. 11. Condino, Francesca, and Filippo Domma. ―The beta-Dagum distribution.‖ 45th Scientific Meeting of the Italian Statistical Society, University of Padua. (2010). 12. Cummins, J. D., Dionne, G., McDonald, J. B., and Pritchett, B. M. (1990). Applications of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics & Economics, 9, 257–272.

262

13. DE MORAIS, ALICE LEMOS. A class of generalized beta distributions, Pareto power series and Weibull power series. Diss. Universidade Federal de Pernambuco, (2009). 14. Dutka, J. (1981). The incomplete beta function—a historical profile. Archive for the History of Exact Sciences, 24, 11–29. 15. Eugene, N., C. Lee, and F. Famoye, (2002). Beta-normal Distribution and its applications. Communication in Statistics – Theory and Methods, 31:497-512. 16. Famoye, F. , Lee, C. & Olugbenga Olumolade, (2005). The Beta-weibull distribution. Journal of Statistical Theory and Applications, 121-138. 17. Famoye, F., Lee, C., and Eugene, N. Beta-Normal Distribution: Bimodality Properties and Application. Journal of Modern Applied Statistical Methods 2 (Nov.2003), 314–326. 18. Fischer, Matthias J.; Vaughan, David (2004) : The Beta-Hyperbolic Secant (BHS) Distribution, Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Statistik und Ökonometrie, No. 64/2004, http://hdl.handle.net/10419/29617 19. Gordy, M.B., (1988a), Computationally convenient distributional assumptions for common-value auctions, Computational Economics 12, 61-78. 20. Gordy, M.B., (1988b), A generalization of generalized beta distributions. Paper No. 1998-18, Finance and Economics Discussion Series. The Federal Reserve Boeard. 21. Grassia, A., (1977), On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions, Australian Journal of Statistics 19, 108-114. 22. Gupta, A. K. and Nadarajah, S. (2004). On the moments of the beta normal distribution. Communication in Statistics – Theory and Methods, 31: 1-13. 23. Gupta, A. and Nadarajah, S. (2004). Handbook of Beta Distribution and Its Applications. New York: Marcel Dekker. 24. Gupta, A. K. and Nadarajah, S.. Beta Bessel Distribution. International Journal of Mathematics and Mathematical Sciences, (2006): 1-14. 25. Gupta, R.D., Kundu, D. (1999). Generalized exponential distributions.

263

Australian and New Zealand Journal of Statistic 41, 173-188 26. J.H. Sepanski and L.Kong, A Family of Generalized Beta Distribution For Income, International Business, 1(10), (2007), 129-145. 27. Jones, M.C. ―Families of distributions arising from distributions of order statistics.‖ Test 13.1 (2004):1-43. 28. Kalbfleisch, J. D., and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. New York: John Wiley. 29. Klugman, S. A., Panjer, H. H., and Willmot, G. E. (1998). Loss Models. New York: John Wiley. 30. Kong, L., Lee, C., and Sepanski, J. H. On the Properties of Beta-Gamma Distribution. Journal of Modern Applied Statistical Methods 6, 1 (2007), 187– 211. 31. L.Cardeńo, D.K Nagar and L.E Sánchez, Beta Type 3 Distribution and Its Multivariate Generalization, Tamsui Oxford Journal of Mathematical Sciences 21(2)(2005).225-241 Aletheia University 32. Lee, C., Famoye, F., and Olumolade, O. Beta-Weibull Distribution: Some Properties and Applications to Censored Data. Journal Of Modern Applied Statistical Methods 6 (May 2007), 173–186. 33. Leemis, L. and Mcqueston, J., Univariate Distribution Relationships The American Statistician, February (2008), vol.62, No.1 45 34. Libby, D.L and Novick, M., (1982), Multivariate generalized beta-distributions with applications to utility assessment. Journal of Educational Statistics, 7, 271- 294. 35. McDonald, J. B., and Ransom, M. (1979a). Functional forms, estimation techniques, and the distribution of income. Econometrica, 47, 1513–1526. 36. McDonald, J. B., and Richards, D. O., (1987a), Model selection: some generalized distributons. Communications in Statistics (volume A), 16, 1049- 1074. 37. McDonald, J. B., and Richards, D. O., (1987b), Some generalized models with applications to reliability. Journal of Statistical Planning and Inference, 16, 365- 376.

264

38. McDonald, J. B., and Y.J.Xu, A Generalization of the Beta Distribution with Application, Journal of Econometrics, 69(2), (1995), 133-152. 39. McDonald, J.B., (1984), Some generalized functions for the size distribution of income. Econometrica, 52, 647-664. 40. Mudholkar, G.S., Srivastava, D.K., Freimer, M. (1995). The exponentiated Weibull family. Technometrics 37 436-445. 41. Nadarajah, S. (2005). Exponentiated beta distributions. Computers &Mathematics with Application. 49, 1029-1035. 42. Nadarajah, S. and Kotz, S., (2004): A generalized beta distribution. The Mathematical Scientist, 29, 36-41. 43. Nadarajah, S., and Gupta, A. K. The Beta Frèchet Distribution. Far East Journal of Theoretical Statistics 14 (2004), 15–24. 44. Nadarajah, S., and Kotz, S. The Beta Exponential Distribution. Reliability Engineering and System Safety 91 (2006), 689-697.

45. Nadarajah, S., and Kotz, S. (2006), An F1 beta distribution with bathtub failure rate function. American Journal of Mathematical and Management Sciences, 26, 113-131. 46. Nadarajah, S., and Kotz, S. (2007): Multitude of beta distributions with applications, Statistics: A Journal of Theoretical and Applied Statistics, 41:2, 153-179 47. Nadarajah, S., and Kotz, S. The Beta Gumbel Distribution. Mathematical Problems in Engineering 4 (2004), 323–332. 48. Paranaiba P.F, Ortega E.M.M, Cordeiro G.M, Pescim R.R.,The beta Burr XII distribution with application to lifetime data, Computational Statistics and Data Analysis (2010) 49. Patil, G.P., M.T.Boswell, and M.V. Ratnaparkhi, (1984), Dictionary and classified bibliography of statistical distributions in scientific work: Continuous univariate models (International cooperative Publishing House, Burtonsville, MD). 50. Pescim,R.R.,Demètrio C.G., Cordeiro, G.M., Ortega, E.M., and Urbano, M.R. (2009). The beta generalized half-normal distribution. Computational Statistics

265

and data analysis, 54(4), 945-957.

51. Thurow, L. C. (1970). Analyzing the American Income Distribution. Papers and proceedings, American Economics Association, 60, 261-269. 52. Vartia, P. L. I., and Vartia, Y. O. (1980). Description of the income distribution by the scaled F distribution model. In: N. A. Klevmarken and J. A. Lybeck (eds.): The Statics and Dynamics of Income. Clevedon, UK: Tieto, pp. 23–36. 53. Venter, G. (1983). Transformed beta and gamma distributions and aggregate losses. Proceedings of the Casualty Actuarial Society, 70, 156–193. 54. Walck, C. (1996). Handbook on Statistical Distributions for Experimetalists. Internal Report SUF–PFY/96–01, Particle Physics Group, Fysikum, Stockholm University, 11 December 1996. 55. Yuan; Ye, Broderick, O. Oluyede; Mavis, Pararai (2012) : Weighted generalized beta distribution of the second kind and related distributions, Journal of Statistical and Econometric Methods, ISSN 2241-0376, International Scientific Press, Vol 1, Iss. 1, 13-31.

266