Univariate Discrete Distributions

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1 Univariate Discrete Distributions

Univariate Discrete Distributions THIRD EDITION NORMAN L. JOHNSON University of North Carolina Department of Statistics Chapel Hill, North Carolina ADRIENNE W. KEMP University of St.

3 Univariate Discrete Distributions THIRD EDITION NORMAN L. JOHNSON University of North Carolina Department of Statistics Chapel Hill, North Carolina ADRIENNE W. KEMP University of St. Andrews Mathematical Institute North Haugh, St. Andrews United Kingdom SAMUEL KOTZ George Washington University Department of Engineering Management and Systems Engineering Washington, D.C. A JOHN WILEY & SONS, INC., PUBLICATION

4 Copyright 2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: ISBN Printed in the United States of America

5 To the memory of Norman Lloyd Johnson ( )

6 Contents Preface xvii 1 Preliminary Information Mathematical Preliminaries, Factorial and Combinatorial Conventions, Gamma and Beta Functions, Finite Difference Calculus, Differential Calculus, Incomplete Gamma and Beta Functions and Other Gamma-Related Functions, Gaussian Hypergeometric Functions, Confluent Hypergeometric Functions (Kummer s Functions), Generalized Hypergeometric Functions, Bernoulli and Euler Numbers and Polynomials, Integral Transforms, Orthogonal Polynomials, Basic Hypergeometric Series, Probability and Statistical Preliminaries, Calculus of Probabilities, Bayes s Theorem, Random Variables, Survival Concepts, Expected Values, Inequalities, Moments and Moment Generating Functions, Cumulants and Cumulant Generating Functions, 54 vii

7 viii CONTENTS Joint Moments and Cumulants, Characteristic Functions, Probability Generating Functions, Order Statistics, Truncation and Censoring, Mixture Distributions, Variance of a Function, Estimation, General Comments on the Computer Generation of Discrete Random Variables, Computer Software, 73 2 Families of Discrete Distributions Lattice Distributions, Power Series Distributions, Generalized Power Series Distributions, Modified Power Series Distributions, Difference-Equation Systems, Katz and Extended Katz Families, Sundt and Jewell Family, Ord s Family, Kemp Families, Generalized Hypergeometric Probability Distributions, Generalized Hypergeometric Factorial Moment Distributions, Distributions Based on Lagrangian Expansions, Gould and Abel Distributions, Factorial Series Distributions, Distributions of Order-k, q-series Distributions, Binomial Distribution Definition, Historical Remarks and Genesis, Moments, Properties, Order Statistics, Approximations, Bounds, and Transformations, 116

8 CONTENTS ix Approximations, Bounds, Transformations, Computation, Tables, and Computer Generation, Computation and Tables, Computer Generation, Estimation, Model Selection, Point Estimation, Confidence Intervals, Model Verification, Characterizations, Applications, Truncated Binomial Distributions, Other Related Distributions, Limiting Forms, Sums and Differences of Binomial-Type Variables, Poissonian Binomial, Lexian, and Coolidge Schemes, Weighted Binomial Distributions, Chain Binomial Models, Correlated Binomial Variables, Poisson Distribution Definition, Historical Remarks and Genesis, Genesis, Poissonian Approximations, Moments, Properties, Approximations, Bounds, and Transformations, Computation, Tables, and Computer Generation, Computation and Tables, Computer Generation, Estimation, Model Selection, Point Estimation, Confidence Intervals, 176

9 x CONTENTS Model Verification, Characterizations, Applications, Truncated and Misrecorded Poisson Distributions, Left Truncation, Right Truncation and Double Truncation, Misrecorded Poisson Distributions, Poisson Stopped Sum Distributions, Other Related Distributions, Normal Distribution, Gamma Distribution, Sums and Differences of Poisson Variates, Hyper-Poisson Distributions, Grouped Poisson Distributions, Heine and Euler Distributions, Intervened Poisson Distributions, Negative Binomial Distribution Definition, Geometric Distribution, Historical Remarks and Genesis of Negative Binomial Distribution, Moments, Properties, Approximations and Transformations, Computation and Tables, Estimation, Model Selection, P Unknown, Both Parameters Unknown, Data Sets with a Common Parameter, Recent Developments, Characterizations, Geometric Distribution, Negative Binomial Distribution, Applications, Truncated Negative Binomial Distributions, Related Distributions, Limiting Forms, 236

10 CONTENTS xi Extended Negative Binomial Model, Lagrangian Generalized Negative Binomial Distribution, Weighted Negative Binomial Distributions, Convolutions Involving Negative Binomial Variates, Pascal Poisson Distribution, Minimum (Riff Shuffle) and Maximum Negative Binomial Distributions, Condensed Negative Binomial Distributions, Other Related Distributions, Hypergeometric Distributions Definition, Historical Remarks and Genesis, Classical Hypergeometric Distribution, Beta Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution, Beta Negative Binomial Distribution: Beta Pascal Distribution, Generalized Waring Distribution, Pólya Distributions, Hypergeometric Distributions in General, Moments, Properties, Approximations and Bounds, Tables, Computation, and Computer Generation, Estimation, Classical Hypergeometric Distribution, Negative (Inverse) Hypergeometric Distribution: Beta Binomial Distribution, Beta Pascal Distribution, Characterizations, Applications, Classical Hypergeometric Distribution, Negative (Inverse) Hypergeometric Distribution: Beta Binomial Distribution, Beta Negative Binomial Distribution: Beta Pascal Distribution, Generalized Waring Distribution, Special Cases, 283

11 xii CONTENTS Discrete Rectangular Distribution, Distribution of Leads in Coin Tossing, Yule Distribution, Waring Distribution, Narayana Distribution, Related Distributions, Extended Hypergeometric Distributions, Generalized Hypergeometric Probability Distributions, Generalized Hypergeometric Factorial Moment Distributions, Other Related Distributions, Logarithmic and Lagrangian Distributions Logarithmic Distribution, Definition, Historical Remarks and Genesis, Moments, Properties, Approximations and Bounds, Computation, Tables, and Computer Generation, Estimation, Characterizations, Applications, Truncated and Modified Logarithmic Distributions, Generalizationsofthe Logarithmic Distribution, Other Related Distributions, Lagrangian Distributions, Otter s Multiplicative Process, Borel Distribution, Consul Distribution, Geeta Distribution, General Lagrangian Distributions of the First Kind, Lagrangian Poisson Distribution, Lagrangian Negative Binomial Distribution, 340

12 CONTENTS xiii Lagrangian Logarithmic Distribution, Lagrangian Distributions of the Second Kind, Mixture Distributions Basic Ideas, Introduction, Finite Mixtures, Varying Parameters, Bayesian Interpretation, Finite Mixtures of Discrete Distributions, Parameters of Finite Mixtures, Parameter Estimation, Zero-Modified and Hurdle Distributions, Examples of Zero-Modified Distributions, Finite Poisson Mixtures, Finite Binomial Mixtures, Other Finite Mixtures of Discrete Distributions, Continuous and Countable Mixtures of Discrete Distributions, Properties of General Mixed Distributions, Properties of Mixed Poisson Distributions, Examples of Poisson Mixtures, Mixtures of Binomial Distributions, Examples of Binomial Mixtures, Other Continuous and Countable Mixtures of Discrete Distributions, Gamma and Beta Mixing Distributions, Stopped-Sum Distributions Generalized and Generalizing Distributions, Damage Processes, Poisson Stopped Sum (Multiple Poisson) Distributions, Hermite Distribution, Poisson Binomial Distribution, Neyman Type A Distribution, Definition, Moment Properties, Tables and Approximations, 406

13 xiv CONTENTS Estimation, Applications, Pólya Aeppli Distribution, Generalized Pólya Aeppli (Poisson Negative Binomial) Distribution, Generalizations of Neyman Type A Distribution, Thomas Distribution, Borel Tanner Distribution: Lagrangian Poisson Distribution, Other Poisson Stopped Sum (multiple Poisson) Distributions, Other Families of Stopped-Sum Distributions, Matching, Occupancy, Runs, and q-series Distributions Introduction, Probabilities of Combined Events, Matching Distributions, Occupancy Distributions, Classical Occupancy and Coupon Collecting, Maxwell Boltzmann, Bose Einstein, and Fermi Dirac Statistics, Specified Occupancy and Grassia Binomial Distributions, Record Value Distributions, Runs Distributions, Runs of Like Elements, Runs Up and Down, Distributions of Order k, Early Work on Success Runs Distributions, Geometric Distribution of Order k, Negative Binomial Distributions of Order k, Poisson and Logarithmic Distributions of Order k, Binomial Distributions of Order k, Further Distributions of Order k, q-series Distributions, Terminating Distributions, q-series Distributions with Infinite Support, Bilateral q-series Distributions, q-series Related Distributions, 476

14 CONTENTS xv 11 Parametric Regression Models and Miscellanea Parametric Regression Models, Introduction, Tweedie Poisson Family, Negative Binomial Regression Models, Poisson Lognormal Model, Poisson Inverse Gaussian (Sichel) Model, Poisson Polynomial Distribution, Weighted Poisson Distributions, Double-Poisson and Double-Binomial Distributions, Simplex Binomial Mixture Model, Miscellaneous Discrete Distributions, Dandekar s Modified Binomial and Poisson Models, Digamma and Trigamma Distributions, Discrete Adès Distribution, Discrete Bessel Distribution, Discrete Mittag Leffler Distribution, Discrete Student s t Distribution, Feller Arley and Gegenbauer Distributions, Gram Charlier Type B Distributions, Interrupted Distributions, Lost-Games Distributions, Luria Delbrück Distribution, Naor s Distribution, Partial-Sums Distributions, Queueing Theory Distributions, Reliability and Survival Distributions, Skellam Haldane Gene Frequency Distribution, Steyn s Two-Parameter Power Series Distributions, Univariate Multinomial-Type Distributions, Urn Models with Stochastic Replacements, Zipf-Related Distributions, Haight s Zeta Distributions, 533 Bibliography 535 Abbreviations 631 Index 633

15 Preface This book is dedicated to the memory of Professor N. L. Johnson, who passed away during the production stages. He was my longtime friend and mentor; his assistance with this revision during his long illness is greatly appreciated. His passing is a sad loss to all who are interested in statistical distribution theory. The preparation of the third edition gave Norman and I the opportunity to substantially revise and reorganize parts of the book. This enabled us to increase the coverage of certain areas and to highlight today s better understanding of interrelationships between distributions. Also a number of errors and inaccuracies in the two previous editions have been corrected and some explanations are clarified. The continuing interest in discrete distributions is evinced by the addition of over 400 new references, nearly all since Electronic databases, such as Statistical Theory and Methods Abstracts (published by the International Statistical Institute), the Current Index to Statistics: Applications, Methods and Theory (published by the American Statistical Association and the Institute of Mathematical Statistics), and the Thomson ISI Web of Science, have drawn to our attention papers and articles which might otherwise have escaped notice. It is important to acknowledge the impact of scholarly, encyclopedic publications such as the Dictionary and Bibliography of Statistical Distributions in Scientific Work, Vol. 1: Discrete Models, by G. P. Patil, M. T. Boswell, S. W. Joshi, and M. V. Ratnaparkhi (1984) (published by the International Co-operative Publishing House, Fairland, MD), and the Thesaurus of Univariate Discrete Probability Distributions, by G. Wimmer and G. Altmann (1999) (published by Stamm Verlag, Essen). The new edition of Statistical Distributions, by M. Evans, N. Peacock, and B. Hastings (2000) (published by Wiley, New York), encouraged us to address the needs of occasional readers as distinct from researchers into the theoretical and applied aspects of the subject. The objectives of this book are far wider. It aims, as before, to give an account of the properties and the uses of discrete distributions at the time of writing, while adhering to the same level and style as previous editions. The 1969 intention to exclude theoretical minutiae of no apparent practical importance has not xvii

16 xviii PREFACE been forgotten. We have tried to give a balanced account of new developments, especially those in the more accessible statistical journals. There has also been relevant work in related fields, such as econometrics, combinatorics, probability theory, stochastic processes, actuarial studies, operational research, and social sciences. We have aimed to provide a framework within which future research findings can best be understood. In trying to keep the book to a reasonable length, some material that should have been included was omitted or its coverage curtailed. Comments and criticisms are welcome; I would like to express our gratitude to friends and colleagues for pointing out faults in the last edition and for their input of ideas into the new edition. The structure of the book is broadly similar to that of the previous edition. The organization of the increased amount of material into the same number of chapters has, however, created some unfamilar bedfellows. An extra chapter would have had an untoward effect on the next two books in the series (Univariate Continuous Distributions, Vols. 1 and 2); these begin with Chapter 12. Concerning numbering conventions, each chapter is divided into sections and within many sections there are subsections. Instead of a separate name index, the listed references end with section numbers enclosed in square brackets. Chapter 1 has seen some reordering and the inclusion of a small amount of new, relevant material. Sections 1.1 and 1.2 contain mathematical preliminaries and statistical preliminaries, respectively. Material on the computer generation of specific types of random variables is shifted to appropriate sections in other chapters. We chose not to discuss software explicitly we felt that this is precluded by shortage of space. Some of the major packages are listed at the end of Chapter 1, however. Many contain modules for tasks associated with specific distributions. Websites are given so that readers can obtain further information. In Chapter 2, most of the material on distributions based on Lagrangian expansions is moved to Chapter 7, which is now entitled Logarithmic and Lagrangian Distributions. There are new short sections in Chapter 2 on order-k and q-series distributions, mentioning their new placement in the book and changes in customary notations since the last edition. Chapters 3, 4, and 5 are structurally little changed, although new sections on chain binomial models (Chapter 3), the intervened Poisson distribution (Chapter 4), and the minimum and maximum negative binomial distributions and the condensed negative binomial distribution (Chapter 5) are added. It is hoped that the limited reordering and insertion of new material in Chapter 6 will improve understanding of hypergeometric-type distributions. Chapter 7 now has a dual role. Logarithmic distributions occupy the first half. The new second part contains a coherent and updated treatment of the previously fragmented material on Lagrangian distributions. The typographical changes in Chapters 8 and 9 are meant to make them more reader friendly. Chapter 10 is now much longer. It contains the section on record value Distributions that was previously in Chapter 11. The treatment of order-k distributions

17 PREFACE xix is augmented by accounts of recent researches. The chapter ends with a consolidated account of the absorption, Euler, and Heine distributions, as well as new q-series material, including new work on the null distribution of the Wilcoxon Mann Whitney test statistic. Chapter 11 has seen most change; it is now in two parts. The ability of modern computers to gather and analyze very large data sets with many covariates has led to the construction of many regression-type models, both parametric and nonparametric. The first part of Chapter 11 gives an account of certain regression models for discrete data that are probabilistically fully specified, that is, fully parametric. These include the Tweedie Poisson family, the Poisson lognormal, Poisson inverse Gaussian, and Poisson polynomial distributions. Efron s double Poisson and double binomial and the simplex-binomial mixture model also receive attention. The remainder of Chapter 11 is on miscellaneous discrete distributions, as before. Those distributions that have fitted better into earlier chapters are replaced with newer ones, such as the discrete Bessel, the discrete Mittag Leffler, and the Luria Delbrück distributions. There is a new section on survival distributions. The section on Zipf and zeta distributions is split into two; renewed interest in the literature in Zipf-type distributions is recognized by the inclusion of Hurwitz zeta and Lerch distributions. We have been particularly indebted to Professors David Kemp and Bala Balakrishnan, who have read the entire manuscript and have made many valuable recommendations (not always implemented). David was particularly helpful with his knowledge of AMS LATEX and his understanding of the Wiley stylefile. He has also been of immense help with the task of proofreading. It is a pleasure to record the facilities and moral support provided by the Mathematical Institute at the University of St Andrews, especially by Dr. Patricia Heggie. Norman and I much regretted that Sam Kotz, with his wide-ranging knowledge of the farther reaches of the subject, felt unable to join us in preparing this new edition. Adrienne W. Kemp St Andrews, Scotland November 2004

Univariate Discrete Distributions

Univariate Discrete Distributions Second Edition NORMAN L. JOHNSON University of North Carolina Chapel Hill, North Carolina SAMUEL KOTZ University of Maryland College Park, Maryland ADRIENNE W. KEMP University