Statistical Distributions in General Insurance Stochastic Processes

Total Page:16

File Type:pdf, Size:1020Kb

Statistical Distributions in General Insurance Stochastic Processes Statistical distributions in general insurance stochastic processes by Jan Hendrik Harm Steenkamp Submitted in partial fulfilment of the requirements for the degree Magister Scientiae In the Department of Statistics In the Faculty of Natural & Agricultural Sciences University of Pretoria Pretoria January 2014 © University of Pretoria 1 I, Jan Hendrik Harm Steenkamp declare that the dissertation, which I hereby submit for the degree Magister Scientiae in Mathematical Statistics at the University of Pretoria, is my own work and has not previously been submit- ted for a degree at this or any other tertiary institution. SIGNATURE: DATE: 31 January 2014 © University of Pretoria Summary A general insurance risk model consists of in initial reserve, the premiums collected, the return on investment of these premiums, the claims frequency and the claims sizes. Except for the initial reserve, these components are all stochastic. The assumption of the distributions of the claims sizes is an integral part of the model and can greatly influence decisions on reinsurance agreements and ruin probabilities. An array of parametric distributions are available for use in describing the distribution of claims. The study is focussed on parametric distributions that have positive skewness and are defined for positive real values. The main properties and parameterizations are studied for a number of distribu- tions. Maximum likelihood estimation and method-of-moments estimation are considered as techniques for fitting these distributions. Multivariate nu- merical maximum likelihood estimation algorithms are proposed together with discussions on the efficiency of each of the estimation algorithms based on simulation exercises. These discussions are accompanied with programs developed in SAS PROC IML that can be used to simulate from the var- ious parametric distributions and to fit these parametric distributions to observed data. The presence of heavy upper tails in the context of general insurance claims size distributions indicates that there exists a high risk of observing very large and even extreme claims. This needs to be allowed for in the modeling of claims. Methods used to describe tail weight together with techniques that can be used to detect the presence of heavy upper tails are studied. These methods are then applied to the parametric distributions to classify their tails' heaviness. The study is concluded with an application of the techniques developed to fit the parametric distributions and to evaluate the tail heaviness of real- life claims data. The goodness-of-fit of the various fitted distributions are discussed. Based on the final results further research topics are identified. 2 © University of Pretoria Acknowledgements To Jesus Christ I am sincerely greatful for being blessed with the opportu- nity to have studied and also for the ability to have conducted this study. I received strength to persist despite setbacks and challenges. To my family and in particular my mother I would like to express my thanks for your support and encouragement throughout the duration of my studies. To my supervisor, Dr. Inger Fabris-Rotelli, thank you for your time, sharing of knowledge, valuable input and guidance during this research. To my employer and colleagues I would like to express my thanks for your support and helping me to free up time to do this research. 3 © University of Pretoria Contents 1 Introduction 18 1.1 Problem Statement . 18 1.2 Literature Review . 21 1.3 Structure of the dissertation . 26 2 General Insurance Overview 28 2.1 Inter-occurrence times . 29 2.2 Number of claims . 30 2.2.1 Poisson Process . 32 2.2.2 Mixed Poisson Process . 34 2.3 Claim size . 37 2.4 Aggregate claim amount . 39 2.5 Premium Income . 42 2.6 Risk Reserve . 43 2.7 Reinsurance . 44 2.8 Ruin Problems . 47 3 Theory and Properties of Distributions 48 3.1 Introduction . 48 3.2 Moments of parametric distributions . 55 3.3 Distributions with monotone hazard rates . 56 3.4 Heavy-tailed distributions . 59 3.4.1 Classification of tail heaviness based on the moment generating function . 59 3.4.2 Classification of tail heaviness based on exponential boundedness . 60 4 © University of Pretoria CONTENTS 5 3.4.3 Comparison of tail weights using limiting tail behaviour 61 3.4.4 Classification of tail heaviness based on the hazard function . 63 3.4.5 Classification of tail heaviness based on the hazard rate function . 65 3.4.6 Classification of tail heaviness based on the mean resid- ual hazard rate function . 66 3.5 Left Censored Variables and Integrated Tail Distributions . 69 3.5.1 Excess Loss Variable . 69 3.5.2 Left Censored . 70 3.5.3 Limited Loss Variable . 70 3.5.4 Coefficient of variation when the distribution is heavy- tailed . 71 3.6 Subexponential Distributions . 74 3.7 Practical Methods to Detect Heavy Tails . 75 3.7.1 Overview . 75 3.7.2 Method 1: Using definition of the subexponential class 75 3.7.3 Method 2: Using the mean residual hazard rate function 76 3.7.4 Method 3: Testing for an exponentially bounded tail using quantile plots . 77 4 Parametric Loss Distributions 80 4.1 Continuous Distributions for Claim Sizes . 81 4.1.1 Gamma Distribution . 81 4.1.2 Exponential Distribution . 81 4.1.3 Chi-square Distribution . 83 4.1.4 Two-parameter Exponential Distribution . 84 4.1.5 Erlang Distribution . 85 4.1.6 Extreme Value Distribution . 87 4.1.7 Frechet Distribution . 92 4.1.8 Weibull Distribution . 93 4.1.9 Rayleigh Distribution . 96 4.1.10 Gumbel Distribution . 96 4.1.11 Pareto (Lomax) Distribution . 100 © University of Pretoria CONTENTS 6 4.1.12 Generalized Pareto Distribution . 102 4.1.13 Lognormal Distribution . 103 4.1.14 Beta-prime Distribution (Pearson Type VI) . 106 4.1.15 Birnbaum-Saunders Distribution . 108 4.1.16 Burr Distribution . 109 4.1.17 Dagum Distribution . 113 4.1.18 Generalized Beta Distribution of the Second Kind . 116 4.1.19 Singh-Maddala Distribution . 118 4.1.20 Kappa Family of Distributions . 121 4.1.21 Loggamma Distribution . 123 4.1.22 Snedecor's F Distribution . 126 4.1.23 Log-logistic Distribution . 128 4.1.24 Folded and Half Normal Distribution . 129 4.1.25 Inverse Gamma Distribution . 133 4.1.26 Inverse Chi-square Distribution . 135 4.1.27 Inverse Gaussian Distribution . 136 4.1.28 Skew-normal Distribution . 138 4.1.29 Exponential-Gamma Distribution . 141 4.2 Other Useful Distributions . 143 4.2.1 Normal Distribution . 143 4.2.2 Logistic Distribution . 146 4.2.3 Dirichlet Distribution . 147 5 Tail Behaviour of Parametric Distributions 149 5.1 Introduction . 149 5.2 Classification of Tail Heaviness for Parametric Continuous Distributions . 150 5.2.1 Gamma and related distributions . 151 5.2.2 Erlang Distribution . 153 5.2.3 Generalized Extreme Value Distribution . 153 5.2.4 Pareto Distribution . 155 5.2.5 Generalized Pareto Distribution . 156 5.2.6 Lognormal Distribution . 158 © University of Pretoria CONTENTS 7 5.2.7 Beta-prime Distribution . 158 5.2.8 Birnbaum-Saunders Distribution . 160 5.2.9 Burr and related distributions . 161 5.2.10 Dagum Distribution . 163 5.2.11 Singh-Maddala Distribution . 164 5.2.12 Kappa Family of Distributions . 165 5.2.13 Generalized Beta Distribution of the Second Kind . 167 5.2.14 Log-logistic Distribution . 168 5.2.15 Folded Normal Distribution . 170 5.2.16 Inverse Gamma and related distributions . 170 5.2.17 Loggamma Distribution . 172 5.2.18 Inverse Gaussian Distribution . 173 5.2.19 Snedecor's F Distribution . 174 5.2.20 Skew-normal Distribution . 174 6 Fitting Parametric Loss Distributions 177 6.1 Introduction . 177 6.2 Numerical Optimisation . 179 6.3 Simulating Parametric Loss Distributions . 180 6.4 Fitting Analysis . 182 6.5 Methods of Fitting Parametric Loss Distributions . 183 6.5.1 Gamma Distribution . 183 6.5.2 Exponential Distribution . 185 6.5.3 Chi-square Distribution . 186 6.5.4 Two-parameter Exponential Distribution . 187 6.5.5 Erlang Distribution . 189 6.5.6 Generalized Extreme Value Distributions . 189 6.5.7 Frechet Distribution . 191 6.5.8 Weibull Distribution . 194 6.5.9 Rayleigh Distribution . 199 6.5.10 Gumbel Distribution . 200 6.5.11 Pareto Distribution . 202 6.5.12 Generalized Pareto Distribution . 203 © University of Pretoria CONTENTS 8 6.5.13 Lognormal Distribution . 205 6.5.14 Beta-prime Distribution . 206 6.5.15 Birnbaum-Saunders Distribution . 208 6.5.16 Burr and related distributions . 209 6.5.17 Dagum Distribution . 210 6.5.18 Generalized Beta Distribution of the Second Kind . 212 6.5.19 Kappa Family of Distributions . 214 6.5.20 Log-logistic Distribution . 216 6.5.21 Folded Normal Distribution . 218 6.5.22 Inverse Gamma and related distributions . 219 6.5.23 Loggamma Distribution . 221 6.5.24 Snedecor's F Distribution . 223 6.5.25 Inverse Gaussian Distribution . 225 6.5.26 Skew-normal Distribution . 227 7 Practical Application 232 7.1 Introduction . 232 7.2 Goodness-of-fit Assessment . 232 7.2.1 Likelihood ratio test . 233 7.2.2 Quantile plots . 234 7.3 Claims Size Data . 236 7.3.1 Background . 236 7.3.2 Description of data and segmentation . 237 7.4 Fitting of Distributions . 238 7.4.1 Approach . 239 7.4.2 Results . 239 7.4.3 Conclusion . 265 8 Conclusion 267 8.1 Overview of study conducted . 267 8.2 Results and Observations . 269 8.3 Future Research . ..
Recommended publications
  • Field Guide to Continuous Probability Distributions
    Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations.
    [Show full text]
  • A Class of Generalized Beta Distributions, Pareto Power Series and Weibull Power Series
    A CLASS OF GENERALIZED BETA DISTRIBUTIONS, PARETO POWER SERIES AND WEIBULL POWER SERIES ALICE LEMOS DE MORAIS Primary advisor: Prof. Audrey Helen M. A. Cysneiros Secondary advisor: Prof. Gauss Moutinho Cordeiro Concentration area: Probability Disserta¸c˜ao submetida como requerimento parcial para obten¸c˜aodo grau de Mestre em Estat´ısticapela Universidade Federal de Pernambuco. Recife, fevereiro de 2009 Morais, Alice Lemos de A class of generalized beta distributions, Pareto power series and Weibull power series / Alice Lemos de Morais - Recife : O Autor, 2009. 103 folhas : il., fig., tab. Dissertação (mestrado) – Universidade Federal de Pernambuco. CCEN. Estatística, 2009. Inclui bibliografia e apêndice. 1. Probabilidade. I. Título. 519.2 CDD (22.ed.) MEI2009-034 Agradecimentos A` minha m˜ae,Marcia, e ao meu pai, Marcos, por serem meus melhores amigos. Agrade¸co pelo apoio `aminha vinda para Recife e pelo apoio financeiro. Agrade¸coaos meus irm˜aos, Daniel, Gabriel e Isadora, bem como a meus primos, Danilo e Vanessa, por divertirem minhas f´erias. A` minha tia Gilma pelo carinho e por pedir `asua amiga, Mirlana, que me acolhesse nos meus primeiros dias no Recife. A` Mirlana por amenizar minha mudan¸cade ambiente fazendo-me sentir em casa durante meus primeiros dias nesta cidade. Agrade¸comuito a ela e a toda sua fam´ılia. A` toda minha fam´ılia e amigos pela despedida de Belo Horizonte e `aS˜aozinha por preparar meu prato predileto, frango ao molho pardo com angu. A` vov´oLuzia e ao vovˆoTunico por toda amizade, carinho e preocupa¸c˜ao, por todo apoio nos meus dias mais dif´ıceis em Recife e pelo apoio financeiro nos momentos que mais precisei.
    [Show full text]
  • Concentration Inequalities from Likelihood Ratio Method
    Concentration Inequalities from Likelihood Ratio Method ∗ Xinjia Chen September 2014 Abstract We explore the applications of our previously established likelihood-ratio method for deriving con- centration inequalities for a wide variety of univariate and multivariate distributions. New concentration inequalities for various distributions are developed without the idea of minimizing moment generating functions. Contents 1 Introduction 3 2 Likelihood Ratio Method 4 2.1 GeneralPrinciple.................................... ...... 4 2.2 Construction of Parameterized Distributions . ............. 5 2.2.1 WeightFunction .................................... .. 5 2.2.2 ParameterRestriction .............................. ..... 6 3 Concentration Inequalities for Univariate Distributions 7 3.1 BetaDistribution.................................... ...... 7 3.2 Beta Negative Binomial Distribution . ........ 7 3.3 Beta-Prime Distribution . ....... 8 arXiv:1409.6276v1 [math.ST] 1 Sep 2014 3.4 BorelDistribution ................................... ...... 8 3.5 ConsulDistribution .................................. ...... 8 3.6 GeetaDistribution ................................... ...... 9 3.7 GumbelDistribution.................................. ...... 9 3.8 InverseGammaDistribution. ........ 9 3.9 Inverse Gaussian Distribution . ......... 10 3.10 Lagrangian Logarithmic Distribution . .......... 10 3.11 Lagrangian Negative Binomial Distribution . .......... 10 3.12 Laplace Distribution . ....... 11 ∗The author is afflicted with the Department of Electrical
    [Show full text]
  • Package 'Extradistr'
    Package ‘extraDistr’ September 7, 2020 Type Package Title Additional Univariate and Multivariate Distributions Version 1.9.1 Date 2020-08-20 Author Tymoteusz Wolodzko Maintainer Tymoteusz Wolodzko <[email protected]> Description Density, distribution function, quantile function and random generation for a number of univariate and multivariate distributions. This package implements the following distributions: Bernoulli, beta-binomial, beta-negative binomial, beta prime, Bhattacharjee, Birnbaum-Saunders, bivariate normal, bivariate Poisson, categorical, Dirichlet, Dirichlet-multinomial, discrete gamma, discrete Laplace, discrete normal, discrete uniform, discrete Weibull, Frechet, gamma-Poisson, generalized extreme value, Gompertz, generalized Pareto, Gumbel, half-Cauchy, half-normal, half-t, Huber density, inverse chi-squared, inverse-gamma, Kumaraswamy, Laplace, location-scale t, logarithmic, Lomax, multivariate hypergeometric, multinomial, negative hypergeometric, non-standard beta, normal mixture, Poisson mixture, Pareto, power, reparametrized beta, Rayleigh, shifted Gompertz, Skellam, slash, triangular, truncated binomial, truncated normal, truncated Poisson, Tukey lambda, Wald, zero-inflated binomial, zero-inflated negative binomial, zero-inflated Poisson. License GPL-2 URL https://github.com/twolodzko/extraDistr BugReports https://github.com/twolodzko/extraDistr/issues Encoding UTF-8 LazyData TRUE Depends R (>= 3.1.0) LinkingTo Rcpp 1 2 R topics documented: Imports Rcpp Suggests testthat, LaplacesDemon, VGAM, evd, hoa,
    [Show full text]
  • Parametric Representation of the Top of Income Distributions: Options, Historical Evidence and Model Selection
    PARAMETRIC REPRESENTATION OF THE TOP OF INCOME DISTRIBUTIONS: OPTIONS, HISTORICAL EVIDENCE AND MODEL SELECTION Vladimir Hlasny Working Paper 90 March, 2020 The CEQ Working Paper Series The CEQ Institute at Tulane University works to reduce inequality and poverty through rigorous tax and benefit incidence analysis and active engagement with the policy community. The studies published in the CEQ Working Paper series are pre-publication versions of peer-reviewed or scholarly articles, book chapters, and reports produced by the Institute. The papers mainly include empirical studies based on the CEQ methodology and theoretical analysis of the impact of fiscal policy on poverty and inequality. The content of the papers published in this series is entirely the responsibility of the author or authors. Although all the results of empirical studies are reviewed according to the protocol of quality control established by the CEQ Institute, the papers are not subject to a formal arbitration process. Moreover, national and international agencies often update their data series, the information included here may be subject to change. For updates, the reader is referred to the CEQ Standard Indicators available online in the CEQ Institute’s website www.commitmentoequity.org/datacenter. The CEQ Working Paper series is possible thanks to the generous support of the Bill & Melinda Gates Foundation. For more information, visit www.commitmentoequity.org. The CEQ logo is a stylized graphical representation of a Lorenz curve for a fairly unequal distribution of income (the bottom part of the C, below the diagonal) and a concentration curve for a very progressive transfer (the top part of the C).
    [Show full text]
  • Probability Distribution Relationships
    STATISTICA, anno LXX, n. 1, 2010 PROBABILITY DISTRIBUTION RELATIONSHIPS Y.H. Abdelkader, Z.A. Al-Marzouq 1. INTRODUCTION In spite of the variety of the probability distributions, many of them are related to each other by different kinds of relationship. Deriving the probability distribu- tion from other probability distributions are useful in different situations, for ex- ample, parameter estimations, simulation, and finding the probability of a certain distribution depends on a table of another distribution. The relationships among the probability distributions could be one of the two classifications: the transfor- mations and limiting distributions. In the transformations, there are three most popular techniques for finding a probability distribution from another one. These three techniques are: 1 - The cumulative distribution function technique 2 - The transformation technique 3 - The moment generating function technique. The main idea of these techniques works as follows: For given functions gin(,XX12 ,...,) X, for ik 1, 2, ..., where the joint dis- tribution of random variables (r.v.’s) XX12, , ..., Xn is given, we define the func- tions YgXXii (12 , , ..., X n ), i 1, 2, ..., k (1) The joint distribution of YY12, , ..., Yn can be determined by one of the suit- able method sated above. In particular, for k 1, we seek the distribution of YgX () (2) For some function g(X ) and a given r.v. X . The equation (1) may be linear or non-linear equation. In the case of linearity, it could be taken the form n YaX ¦ ii (3) i 1 42 Y.H. Abdelkader, Z.A. Al-Marzouq Many distributions, for this linear transformation, give the same distributions for different values for ai such as: normal, gamma, chi-square and Cauchy for continuous distributions and Poisson, binomial, negative binomial for discrete distributions as indicated in the Figures by double rectangles.
    [Show full text]
  • Distributions.Jl Documentation Release 0.6.3
    Distributions.jl Documentation Release 0.6.3 JuliaStats January 06, 2017 Contents 1 Getting Started 3 1.1 Installation................................................3 1.2 Starting With a Normal Distribution...................................3 1.3 Using Other Distributions........................................4 1.4 Estimate the Parameters.........................................4 2 Type Hierarchy 5 2.1 Sampleable................................................5 2.2 Distributions...............................................6 3 Univariate Distributions 9 3.1 Univariate Continuous Distributions...................................9 3.2 Univariate Discrete Distributions.................................... 18 3.3 Common Interface............................................ 22 4 Truncated Distributions 27 4.1 Truncated Normal Distribution...................................... 28 5 Multivariate Distributions 29 5.1 Common Interface............................................ 29 5.2 Multinomial Distribution......................................... 30 5.3 Multivariate Normal Distribution.................................... 31 5.4 Multivariate Lognormal Distribution.................................. 33 5.5 Dirichlet Distribution........................................... 35 6 Matrix-variate Distributions 37 6.1 Common Interface............................................ 37 6.2 Wishart Distribution........................................... 37 6.3 Inverse-Wishart Distribution....................................... 37 7 Mixture Models 39 7.1 Type
    [Show full text]
  • Skewed Reflected Distributions Generated by the Laplace Kernel 1 Introduction
    AUSTRIAN JOURNAL OF STATISTICS Volume 38 (2009), Number 1, 45–58 Skewed Reflected Distributions Generated by the Laplace Kernel M. Masoom Ali1, Manisha Pal2 and Jungsoo Woo3 1Dept. of Mathematical Sciences, Ball State University, Indiana, USA 2Dept. of Statistics, University of Calcutta, India 3Dept. of Statistics, Yeungnam University, Gyongsan, South Korea Abstract: In this paper we construct some skewed distributions with pdfs of the form 2f(u)G(¸u), where ¸ is a real number, f(¢) is taken to be a Laplace pdf while the cdf G(¢) comes from one of Laplace, double Weibull, reflected Pareto, reflected beta prime, or reflected generalized uniform distribution. Properties of the resulting distributions are studied. In particular, expres- sions for the moments of these distributions and the characteristic functions are derived. However, as some of these quantities could not be evaluated in closed forms, special functions have been used to express them. Graphical illustrations of the pdfs of the skewed distributions are also given. Further, skewness-kurtosis graphs for these distributions have been drawn. Zusammenfassung: In diesem Aufsatz konstruieren wir einige schiefe Vertei- lungen mit Verteilungsfunktionen der Form 2f(u)G(¸u), wobei ¸ eine reelle Zahl und f(¢) eine Laplace Dichte ist, wahrend¨ die Verteilungsfunktion G(¢) von einer Laplace-, zweiseitigen Weibull-, reflektierten Pareto-, reflektierten Beta-Primar-,¨ oder reflektierten generalisierten Gleichverteilung stammt. Die Eigenschaften der resultierenden Verteilungen werden untersucht. Im Beson- deren werden Ausdrucke¨ fur¨ die Momente dieser Verteilungen und deren charakteristische Funktionen hergeleitet. Da jedoch einige dieser Großen¨ nicht in geschlossener Form dargestellt werden konnten, wurden spezielle Funktionen verwendet um diese auszudrucken.¨ Graphische Illustrationen der Dichten dieser schiefen Verteilungen sind auch gegeben.
    [Show full text]
  • Combinatorial Clustering and the Beta Negative Binomial Process
    1 Combinatorial Clustering and the Beta Negative Binomial Process Tamara Broderick, Lester Mackey, John Paisley, Michael I. Jordan Abstract We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process (NBP) as an infinite- dimensional prior appropriate for such problems. We show that the NBP is conjugate to the beta process, and we characterize the posterior distribution under the beta-negative binomial process (BNBP) and hierarchical models based on the BNBP (the HBNBP). We study the asymptotic properties of the BNBP and develop a three-parameter extension of the BNBP that exhibits power-law behavior. We derive MCMC algorithms for posterior inference under the HBNBP, and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis. ✦ 1 INTRODUCTION In traditional clustering problems the goal is to induce a set of latent classes and to assign each data point to one and only one class. This problem has been approached within a model-based framework via the use of finite mixture models, where the mixture components characterize the distributions associated with the classes, and the mixing proportions capture the mutual exclusivity of the classes (Fraley and Raftery, 2002; arXiv:1111.1802v5 [stat.ME] 10 Jun 2013 McLachlan and Basford, 1988). In many domains in which the notion of latent classes is natural, however, it is unrealistic to assign each individual to a single class.
    [Show full text]
  • From the Classical Beta Distribution to Generalized Beta Distributions
    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 ii 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. iv 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.
    [Show full text]
  • Probability Distributions
    Probability distributions [Nematrian website page: ProbabilityDistributionsIntro, © Nematrian 2020] The Nematrian website contains information and analytics on a wide range of probability distributions, including: Discrete (univariate) distributions - Bernoulli, see also binomial distribution - Binomial - Geometric, see also negative binomial distribution - Hypergeometric - Logarithmic - Negative binomial - Poisson - Uniform (discrete) Continuous (univariate) distributions - Beta - Beta prime - Burr - Cauchy - Chi-squared - Dagum - Degenerate - Error function - Exponential - F - Fatigue, also known as the Birnbaum-Saunders distribution - Fréchet, see also generalised extreme value (GEV) distribution - Gamma - Generalised extreme value (GEV) - Generalised gamma - Generalised inverse Gaussian - Generalised Pareto (GDP) - Gumbel, see also generalised extreme value (GEV) distribution - Hyperbolic secant - Inverse gamma - Inverse Gaussian - Johnson SU - Kumaraswamy - Laplace - Lévy - Logistic - Log-logistic - Lognormal - Nakagami - Non-central chi-squared - Non-central t - Normal - Pareto - Power function - Rayleigh - Reciprocal - Rice - Student’s t - Triangular - Uniform - Weibull, see also generalised extreme value (GEV) distribution Continuous multivariate distributions - Inverse Wishart Copulas (a copula is a special type of continuous multivariate distribution) - Clayton - Comonotonicity - Countermonotonicity (only valid for 푛 = 2, where 푛 is the dimension of the input) - Frank - Generalised Clayton - Gumbel - Gaussian - Independence
    [Show full text]
  • The Exact Distribution of the Ratio of Two Independent Hypoexponential Random Variables Abstract 1 Introduction
    British Journal of Mathematics & Computer Science X(X): XX-XX, 20XX SCIENCEDOMAIN international www.sciencedomain.org The Exact Distribution of The Ratio of Two Independent Hypoexponential Random Variables Therrar Kadri∗1 and Khaled Smaili2 1Department of Mathematics, Faculty of Sciences, Beirut Arab University, Beirut, Lebanon, 2Department of Applied Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon Original Research Article Received: XX December 20XX Accepted: XX December 20XX Online Ready: XX December 20XX Abstract The distribution of ratio of two random variables are of interest in many areas of the sciences. The study of the ratio of same family and finding their exact density expression was examined by many authors for decades. Some are solved and others gave approximations, but many still unsolved. In this paper, we consider the ratio of two independent Hypoexponential distributions. We find the exact expressions for the probability density function, the cumulative distribution function, moment generating function, the reliability function and hazard function, which was proved to be a linear combination of the Generalized-F distribution. Keywords: Ratio Distribution, Hypoexponential Distribution; Erlang Distribution; Generalized-F Distribution; Log-logistic Distribution; Probability Density Function; Cumulative Distribution Function; Reliability Function; Hazard Function. 1 Introduction The distributions of ratio of random variables are widely used in many applied problems of engineering, physics, number theory, order statistics, economics, biology, genetics, medicine, hydrology, psychology, classification, and ranking and selection, see [1,2,3,4]. Examples include safety factor in engineering, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, inventory ratios in economics and Mendelian inheritance ratios in genetics, see [1,2].
    [Show full text]