Prem C. Consul Felix Famoye

Lagrangian Probability Distributions

Birkhäuser Boston • Basel • Berlin Contents

Foreword vii

Preface ix

List of Tables xviii

Abbreviations xix

1 Preliminary Information 1 1.1 Introduction 1 1.2 Mathematical Symbols and Results 1 1.2.1 Combinatorial and Factorial Symbols 1 1.2.2 Gamma and Beta Functions 4 1.2.3 Difference and Differential Calculus 5 1.2.4 Stirling Numbers 7 1.2.5 Hypergeometric Functions 9 1.2.6 Lagrange Expansions 10 1.2.7 Abel and Gould Series 13 1.2.8 Faä di Bruno's Formula 13 1.3 Probabilistic and Statistical Results 14 1.3.1 Probabilities and Random Variables 14 1.3.2 Expected Values 15 1.3.3 Moments and Moment Generating Functions 16 1.3.4 Cumulants and Cumulant Generating Functions 18 1.3.5 Probability Generating Functions ..." 18 1.3.6 Inference 19

2 Lagrangian Probability Distributions 21 2.1 Introduction 21 2.2 Lagrangian Probability Distributions 22 2.2.1 Equivalence of the Two Classes of Lagrangian Distributions 30 2.2.2 Moments of Lagrangian Distributions 33 Contents 2.2.3 Applications of the Results on Mean and Variance 36 2.2.4 Convolution Property for Lagrangian Distributions 36 2.2.5 Probabilistic Structure of Lagrangian Distributions L(/; g; x) 39 2.3 Modified Power Series Distributions 41 2.3.1 Modified Power Series Based on Lagrange Expansions 42 2.3.2 MPSD as a Subclass of Lagrangian Distributions L(f; g\ x) 42 2.3.3 Mean and Variance of a MPSD 45 2.3.4 Maximum Entropy Characterization of some MPSDs 46 2.4 Exercises 48

Properties of General Lagrangian Distributions 51 3.1 Introduction 51 3.2 Central Moments of Lagrangian Distribution L(/; g; x) 51 3.3 Central Moments of Lagrangian Distribution L\ (/r, g; y) 56 3.3.1 Cumulants of Lagrangian Distribution L\{f\\ g; y) 60 3.3.2 Applications 61 3.4 Relations between the Two Classes L(/; g; x) and Li(/r, g; y) 62 3.5 Some Limit Theorems for Lagrangian Distributions 66 3.6 Exercises 67

Quasi-Probability Models 69 4.1 Introduction 69 4.2 Quasi-Binomial Distribution I (QBD-I) 70 4.2.1 QBD-I as a True Probability Distribution 70 4.2.2 Mean and Variance of QBD-I 71 4.2.3 Negative Moments of QBD-I 73 4.2.4 QBD-I Model Based on Difference-Differential Equations 75 4.2.5 Maximum Likelihood Estimation 77 4.3 Quasi-Hypergeometric Distribution I 80 4.4 Quasi-Pölya Distribution I 81 4.5 Quasi-Binomial Distribution II 82 4.5.1 QBD-II as a True Probability Model 83 4.5.2 Mean and Variance of QBD-II 83 4.5.3 Some Other Properties of QBD-II 85 4.6 Quasi-Hypergeometric Distribution II 85 4.7 Quasi-Pölya Distribution II (QPD-II) 86 4.7.1 Special and Limiting Cases 87 4.7.2 Mean and Variance of QPD-II 88 4.7.3 Estimation of Parameters of QP-D-II 88 4.8 Gould Series Distributions 89 4.9 Abel Series Distributions 90 4.10 Exercises 90 Contents xiii 5 Some Urn Models 93 5.1 Introduction 93 5.2 A Generalized Stochastic Urn Model 94 5.2.1 Some Interrelations among Probabilities 100 5.2.2 Recurrence Relation for Moments 101 5.2.3 Some Applications of Prem Model 103 5.3 Urn Model with Predetermined Strategy for Quasi-Binomial Distribution I .. 104 5.3.1 Sampling without Replacement from Urn B 104 5.3.2 Pölya-type Sampling from Urn B 105 5.4 Urn Model with Predetermined Strategy for Quasi-Pölya Distribution II 105 5.4.1 Sampling with Replacement from Urn D 106 5.4.2 Sampling without Replacement from Urn D 107 5.4.3 Urn Model with Inverse Sampling 107 5.5 Exercises 108

6 Development of Models and Applications 109 6.1 Introduction 109 6.2 Branching Process 109 6.3 Queuing Process 111 6.3.1 G|D|1 Queue 112 6.3.2 M|G|1 Queue 113 6.4 Stochastic Model of Epidemics 115 6.5 Enumeration of Trees 116 6.6 Cascade Process 117 6.7 Exercises 118

7 Modified Power Series Distributions 121 7.1 Introduction 121 7.2 Generating Functions 122 7.3 Moments, Cumulants, and Recurrence Relations 122 7.4 Other Interesting Properties 125 7.5 Estimation 128 7.5.1 Maximum Likelihood Estimation of 6 128 7.5.2 Minimum Variance Unbiased Estimation 130 7.5.3 Interval Estimation 132 7.6 Some Characterizations 132 7.7 Related Distributions 136 7.7.1 Inflated MPSD 136 7.7.2 Left Truncated MPSD ', 137 7.8 Exercises 140

8 Some Basic Lagrangian Distributions 143 8.1 Introduction 143 8.2 Geeta Distribution 143 8.2.1 Definition 143 8.2.2 Generating Functions 144 Contents 8.2.3 Moments and Recurrence Relations 145 8.2.4 Other Interesting Properties 145 8.2.5 Physical Models Leading to Geeta Distribution 146 8.2.6 Estimation 148 8.2.7 Some Applications 150 8.3 Consul Distribution 151 8.3.1 Definition 151 8.3.2 Generating Functions 152 8.3.3 Moments and Recurrence Relations 152 8.3.4 Other Interesting Properties 153 8.3.5 Estimation 154 8.3.6 Some Applications 157 8.4 Borel Distribution 158 8.4.1 Definition 158 8.4.2 Generating Functions 158 8.4.3 Moments and Recurrence Relations 159 8.4.4 Other Interesting Properties 159 8.4.5 Estimation 160 8.5 Weighted Basic Lagrangian Distributions 161 8.6 Exercises 162

Generalized Poisson Distribution 165 9.1 Introduction and Definition 165 9.2 Generating Functions 166 9.3 Moments, Cumulants, and Recurrence Relations 167 9.4 Physical Models Leading to GPD 167 9.5 Other Interesting Properties 170 9.6 Estimation 170 9.6.1 Point Estimation 170 9.6.2 Interval Estimation 173 9.6.3 Confidence Regions 175 9.7 Statistical Testing 175 9.7.1 Test about Parameters 175 9.7.2 Chi-Square Test 176 9.7.3 Empirical Distribution Function Test 177 9.8 Characterizations 177 9.9 Applications 179 9.10 Truncated Generalized Poisson Distribution 180 9.11 Restricted Generalized Poisson Distribution .. /. 182 9.11.1 Introduction and Definition 182 9.11.2 Estimation 182 9.11.3 Hypothesis Testing 185 9.12 Other Related Distributions 186 9.12.1 Compound and Weighted GPD 186 9.12.2 Differences of Two GP Variates 187 9.12.3 Absolute Difference of Two GP Variates 188 9.12.4 Distribution of Order Statistics when Sample Size Is a GP Variate ... 188 Contents xv 9.12.5 The Normal and Inverse Gaussian Distributions 189 9.13 Exercises 189

10 Generalized Negative Binomial Distribution 191 10.1 Introduction and Definition 191 10.2 Generating Functions 192 10.3 Moments, Cumulants, and Recurrence Relations 192 10.4 Physical Models Leading to GNBD 194 10.5 Other Interesting Properties 197 10.6 Estimation 200 10.6.1 Point Estimation 201 10.6.2 Interval Estimation 204 10.7 Statistical Testing 205 10.8 Characterizations 207 10.9 Applications 215 10.10 Truncated Generalized Negative Binomial Distribution 217 10.11 Other Related Distributions 218 10.11.1 Poisson-Type Approximation 218 10.11.2 Generalized Logarithmic Series Distribution-Type Limit 218 10.11.3 Differences of Two GNB Variates 219 10.11.4 Weighted Generalized Negative Binomial Distribution 220 10.12 Exercises 220

11 Generalized Logarithmic Series Distribution 223 11.1 Introduction and Definition 223 11.2 Generating Functions 224 11.3 Moments, Cumulants, and Recurrence Relations 225 11.4 Other Interesting Properties 226 11.5 Estimation 229 11.5.1 Point Estimation 229 11.5.2 Interval Estimation 233 11.6 Statistical Testing 233 11.7 Characterizations 234 11.8 Applications 236 11.9 Related Distributions 236 11.10 Exercises 238

12 Lagrangian Katz Distribution 241 12.1 Introduction and Definition '. 241 12.2 Generating Functions 241 12.3 Moments, Cumulants, and Recurrence Relations 242 12.4 Other Important Properties 244 12.5 Estimation 245 12.6 Applications 248 12.7 Related Distributions 248 12.7.1 Basic LKD of Type 1 248 xvi Contents

12.7.2 Basic LKD of Type II 249 12.7.3 Basic GLKD of Type II 250 12.8 Exercises 251

13 Random Walks and Jump Models 253 13.1 Introduction 253 13.2 Simplest Random Walk with Absorbing Barrier at the Origin 254 13.3 Gambler's Ruin Random Walk 254 13.4 Generating Function of Ruin Probabilities in a Polynomial Random Walk ... 255 13.5 Trinomial Random Walks 259 13.6 Quadrinomial Random Walks 260 13.7 Binomial Random Walk (Jumps) Model 262 13.8 Polynomial Random Jumps Model 263 13.9 General Random Jumps Model 264 13.10 Applications 265 13.11 Exercises 266

14 Bivariate Lagrangian Distributions 269 14.1 Definitions and Generating Functions 269 14.2 Cumulants of Bivariate Lagrangian Distributions 271 14.3 Bivariate Modified Power Series Distributions 272 14.3.1 Introduction 272 14.3.2 Moments of BMPSD 273 14.3.3 Properties of BMPSD 275 14.3.4 Estimation of BMPSD 276 14.4 Some Bivariate Lagrangian Delta Distributions 280 14.5 Bivariate Lagrangian Poisson Distribution 281 14.5.1 Introduction 281 14.5.2 Moments and Properties 282 14.5.3 Special BLPD 283 14.6 Other Bivariate Lagrangian Distributions 283 14.6.1 Bivariate Lagrangian Binomial Distribution 283 14.6.2 Bivariate Lagrangian Negative Binomial Distribution 284 14.6.3 Bivariate Lagrangian Logarithmic Series Distribution 286 14.6.4 Bivariate Lagrangian Borel-Tanner Distribution 287 14.6.5 Bivariate Inverse Trinomial Distribution 288 14.6.6 Bivariate Quasi-Binomial Distribution 289 14.7 Exercises 290

15 Multivariate Lagrangian Distributions 293 15.1 Introduction 293 15.2 Notation and Multivariate Lagrangian Distributions 293 15.3 Means and Variance-Covariance 297 15.4 Multivariate Lagrangian Distributions (Special Form) 299 15.4.1 Multivariate Lagrangian Poisson Distribution 299 15.4.2 Multivariate Lagrangian Negative Binomial Distribution 300 Contents xvii

15.4.3 Multivariate Lagrangian Logarithmic Series Distribution 300 15.4.4 Multivariate Lagrangian Delta Distributions 301 15.5 Multivariate Modified Power Series Distributions 302 15.5.1 Multivariate Lagrangian Poisson Distribution 303 15.5.2 Multivariate Lagrangian Negative Binomial Distribution 304 15.5.3 Multivariate Lagrangian Logarithmic Series Distribution 304 15.5.4 Moments of the General Multivariate MPSD 304 15.5.5 Moments of Multivariate Lagrangian Poisson Distribution 305 15.5.6 Moments of Multivariate Lagrangian Negative Binomial Distribution 307 15.6 Multivariate MPSDs in Another Form 308 15.7 Multivariate Lagrangian Quasi-Pölya Distribution 312 15.8 Applications of Multivariate Lagrangian Distributions 312 15.8.1 Queuing Processes 313 15.8.2 Random Branching Processes with k Types of Females 314 15.9 Exercises 315

16 Computer Generation of Lagrangian Variables 317 16.1 Introduction and Generation Procedures 317 16.1.1 Inversion Method 317 16.1.2 Alias Method 318 16.2 Basic Lagrangian Random Variables 319 16.3 Simple Delta Lagrangian Random Variables 321 16.4 Generalized Poisson Random Variables 324 16.5 Generalized Negative Binomial Random Variables 326 16.6 Generalized Logarithmic Series Random Variables 330 16.7 Some Quasi-Type Random Variables 334

References 337

Index 347