Bivariate Discrete Distributions

by Kocherlakota

• ISBN13: 9780824787028
• ISBN10: 0824787021
• eBook ISBN(s): 9781351463454
• Additional ISBN(s): 9781351463454
• Edition: 1st
• Format: Hardcover
• Copyright: 1992-05-18
• Publisher: CRC Press

Summary

This useful reference/text provides a comprehensive study of the various bivariate discrete distributions that have appeared in the literature -- written in an accessible manner that assumes no more than a first course in mathematical statistics. Supplying individualized treatment of topics while simultaneously exploiting the interrelationships of the material, Bivariate Discrete Distributions details the latest techniques of computer simulation for the distributions considered ... contains a general introduction to the structural properties of discrete distributions, including generating functions, moment relationships, and the basic ideas of generalizing ... develops distributions using sampling schemes ... explores the role of compounding ... covers Waring and "short" distributions for use in accident theory ... discusses problems of statistical inference, emphasizing techniques pertinent to the discrete case ... and much more! Containing over 1000 helpful equations, Bivariate Discrete Distributions is an essential, self-contained reference for applied and general statisticians, and biostatisticians; biometricians and mathematicians; and researchers in actuarial science, accident theory, environmental science, ecology, and the study of wildlife populations; and an outstanding text for upper-level undergraduate and graduate students in these disciplines. Book jacket.

Author Biography

Subrahmaniam Kocherlakota is a Professor in the Department of Statistics at the University of Manitoba, Winnipeg, Canada Kathleen Kocherlakota is a Professor in the Department of Statistics at the University of Manitoba, Winnipeg, Canada

Table of Contents

Preface	p. v
Notation	p. xv
Preliminaries	p. 1
Generating functions	p. 2
Convolutions	p. 8
Marginal and conditional distributions	p. 12
Sum and difference of the random variables	p. 15
Conditional distribution of X given the sum	p. 15
Conditional distribution of X given the difference	p. 17
Homogeneous probability generating function	p. 18
Non-homogeneous probability generating functions	p. 20
Compounding and generalizing	p. 23
Structure of bivariate distributions: polynomial expansions	p. 25
Canonical variables	p. 25
Polynomial representation	p. 27
Computer simulation	p. 29
Conditional distribution technique	p. 30
Convolutions	p. 31
Mixtures	p. 31
Trivariate reduction	p. 32
Statistical Inference	p. 33
Estimation	p. 33
Method of moments	p. 34
Method of even-points	p. 40
Zero-zero cell frequency technique	p. 42
Method of maximum likelihood	p. 43
Tests of hypotheses	p. 47
Tests for goodness-of-fit	p. 47
Tests for composite hypotheses	p. 49
Sampling with Replacement	p. 53
Bivariate Bernoulli trials	p. 53
Bivariate Bernoulli distribution	p. 56
Type I bivariate binomial distribution	p. 57
Type II bivariate binomial distribution	p. 63
Trinomial distribution	p. 69
Canonical representation	p. 71
Estimation of the parameters	p. 76
Trinomial distribution	p. 76
Type I bivariate binomial distribution	p. 79
Bivariate Poisson Distribution	p. 87
Introduction	p. 87
Models	p. 87
Properties of the distribution	p. 90
Related distributions	p. 95
Polynomial representation	p. 97
A note on the correlation: Infinite divisibility	p. 99
Estimation	p. 101
Complete distribution	p. 102
Truncated distribution	p. 111
Tests of hypotheses	p. 113
Tests for goodness-of-fit	p. 114
Tests of independence	p. 116
Computer generation of the distribution	p. 118
Bivariate Negative Binomial Distribution	p. 121
Introduction	p. 121
Inverse sampling	p. 122
Bivariate shock model	p. 124
Compounding	p. 127
Canonical representation	p. 133
Mitchell and Paulson model	p. 137
Bivariate geometric distribution	p. 138
Bivariate negative binomial distribution	p. 142
Marshall-Olkin model	p. 144
Limiting forms	p. 145
Estimation	p. 147
p[subscript 3] = 0	p. 147
p[subscript 3 not equal] 0	p. 148
Testing the model	p. 154
Simulation	p. 156
Sampling from Finite Populations	p. 159
Introduction	p. 159
Bivariate generalizations	p. 159
Double dichotomy	p. 160
Trichotomous populations	p. 161
Comparison of the models	p. 162
Marginal and conditional distributions	p. 163
Wicksell form	p. 164
Bivariate hypergeometric distribution	p. 166
Probability generating functions	p. 168
Models for the bivariate hypergeometric distribution	p. 171
Related models based on sampling without replacement	p. 173
Probability distributions	p. 174
Unified model	p. 184
Estimation in the hypergeometric distributions	p. 187
Completeness and sufficiency	p. 188
Maximum likelihood estimation	p. 188
Bivariate Logarithmic Series Distribution	p. 191
Construction of the distribution	p. 191
Properties of the distribution	p. 192
Sum and difference	p. 199
[theta subscript 3] = 0	p. 200
[theta subscript 3 not equal] 0	p. 203
Estimation	p. 208
[theta subscript 3] = 0	p. 208
[theta subscript 3]/[theta subscript 1 theta subscript 2] = [rho] is known	p. 212
[theta subscript 3 not equal] 0	p. 213
Tests of hypotheses	p. 216
Goodness-of-fit	p. 216
Test of the hypothesis [theta subscript 3] = 0	p. 218
Other models for the bivariate LSD	p. 220
Mixture models	p. 221
Model for a modified LSD	p. 224
Computer generation of the distribution	p. 225
Compounded Bivariate Poisson Distributions	p. 227
Introduction	p. 227
Bivariate Poisson: General results	p. 227
Some properties of the compound Poisson	p. 230
Bivariate Neyman type A	p. 233
Properties of the distribution	p. 234
Estimation	p. 238
Computer simulation	p. 244
Bivariate Hermite distribution	p. 245
Genesis of the bivariate Hermite distribution	p. 246
Properties of the distribution	p. 248
Estimation	p. 253
Computer simulation	p. 257
Inverse Gaussian-bivariate Poisson distribution	p. 258
Some properties of the distribution of [gamma] = -1/2	p. 259
Conditional distributions and regression	p. 265
Estimation	p. 267
Simulation	p. 269
Some Miscellaneous Results	p. 271
Introduction	p. 271
Bivariate Waring distribution	p. 271
Properties of the bivariate Waring distribution	p. 274
Estimation	p. 278
Models leading to the BVWD	p. 280
Limiting distributions	p. 284
'Short' distributions	p. 285
Properties of the BVSD	p. 287
Estimation and fitting of the distribution	p. 291
Power series type distributions	p. 295
Bivariate GPSD	p. 296
Mixtures of GPSD: Bates-Neyman model	p. 302
Bibliography	p. 307
Key Word Index	p. 351
Subject Index	p. 359
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