Discrete Multivariate Distributions
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1 Discrete Multivariate Distributions NORMAN L. JOHNSON University of North Carolina Chapel Hill, North Carolina SAMUEL KOTZ University of Maryland College Park, Maryland N. BALAKRISHNAN McMaster University Hamilton, Ontario, Canada A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto
2 Contents Preface List of Tables Glossary Notations xiii xviii xix xxi 34. General Remarks 1 1. Introduction, 1 2. Features of Multivariate Distributions, Probabilities and Probability Generating Function, Moments and Moment-Type Generating Functions, Stirling Numbers and Moments, Recurrence Relations, Incomplete Moments, Conditional Distributions and Moments, Discrete Multivariate Families, Multivariate Discrete Exponential Family, Multivariate Lagrangian Distributions, Homogeneous Multivariate Distributions, Mixed, Modified, and Truncated Distributions, Mixture Distributions, "Inflated" Distributions, Truncated Distributions, Approximations, Product Approximations, Estimation and Structural Approximations, Log-Concavity of Discrete Multivariate Distributions, 27 Bibliography, 29 vii
3 VIII CONTENTS 35. Multinomial Distributions Introduction, Definition and Genesis, 31 3., Moments and Other Properties, Calculation of Sums of Multinomial Probabilities, Approximations, Estimation, Characterizations, Random Number Generation, Applications of Multinomial Distributions, Incomplete and Modified Multinomial Distributions, Truncated Multinomial Distributions, Mixtures of Multinomials, Compound Multinomial and Markov Multinomial Distributions, Compound Multinomial, Markov Multinomial, 83 Bibliography, Negative Multinomial and Other Multinomial-Related Distributions Definition, Historical Remarks and Applications, Properties, Moments and Cumulants, Estimation, Dirichlet (Beta-) Compound Negative Multinomial Distribution, Multivariate Bernoulli, Binomial, and Multinomial Distributions, Countable Mixtures of Bivariate Binomial Distributions, Multivariate Geometric Distributions of Order s, 117 Bibliography, Multivariate Poisson Distributions Introduction, Bivariate Poisson Distributions, Distributions Related to Bivariate Poisson, Multivariate Forms, 139 Bibliography, 147
4 CONTENTS IX 38. Multivariate Power Series Distributions Introduction, Moments and Other Properties, Estimation, Multivariate Logarithmic Series Distributions, Mixtures of Multivariate Power Series Distributions, Multivariate Sum-Symmetric Power Series Distributions, Equivalent Families of Distributions, 167 Bibliography, Multivariate Hypergeometric and Related Distributions Introduction and Genesis, Distribution and Moments, Truncated Multivariate Hypergeometric Distributions, Related Distributions, Multivariate Inverse Hypergeometric and Negative Hypergeometric Distributions, Multivariate Generalized Hypergeometric and Unified Multivariate Hypergeometric Distributions, Restricted Occupancy Distributions (Equiprobable Multivariate Hypergeometric Distributions), "Noncentral" Multivariate Hypergeometric Distributions, Multivariate Hypergeometric-Multinomial Model, Multivariate Lagrange Distributions, Iterated Hypergeometric Distributions, Generalized Waring Distributions, Applications to Faulty Inspection, 193 Bibliography, Multivariate Polya-Eggenberger Distributions Genesis, Distribution, Moments and Mode, Generalizations, Estimation, Special Cases, Polytomic Multivariate P61ya Distributions, Truncated Multivariate P61ya-Eggenberger Distributions, 217
5 X CONTENTS 9. Multivariate Inverse Polya-Eggenberger Distributions, Multivariate Modified Polya Distributions, Genesis, Properties, Multivariate Modified Inverse Polya Distributions, Genesis, Properties, 227 Bibliography, Multivariate Ewens Distribution Genesis and History, Distribution, Moments, and Structural Properties, Moments, Urn Models, Species Deletion (Noninterference), Characterizations, Estimation, Relations with Other Distributions, The GEM Distribution, The Pitman Sampling Formula, Approximations, Applications, Genetics, Bayesian Statistics, Permutations, Ecology, Physics, The Spread of News and Rumors, The Law of Succession, Prime Numbers, Random Mappings, Combinatorial Structures, 242 Bibliography, Multivariate Distributions of Order s Multivariate Geometric Distributions of Order s, Multivariate Distributions in Correlated Binary Variables, Markov Dependence, 251
6 CONTENTS Xi 2.2. Binary Sequences of Order s, Multivariate Negative Binomial Distributions of Order s, Multivariate Poisson Distributions of Order s, Multivariate Logarithmic Series Distributions of Order s, Multinomial Distributions of Order s, Multivariate Polya and Inverse Polya Distributions of Order s, 263 Bibliography, Miscellaneous Distributions Multivariate Occupancy Distributions, Introduction, Distributions and Moments, Multivariate Mixture Distributions, Generalizations, Multivariate Generalized (Random Sum) Distributions, Multivariate Weighted Distributions, Discrete Dirichlet Distributions, Discrete Normal Distributions, 279 Bibliography, 281 Abbreviations 283 Author Index 285 Subject Index 293
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