Preface xvii
1 Preliminary Information 1
1.1 Mathematical Preliminaries 1
1.1.1 Factorial and Combinatorial Conventions 1
1.1.2 Gamma and Beta Functions 5
1.1.3 Finite Difference Calculus 10
1.1.4 Differential Calculus 14
1.1.5 Incomplete Gamma and Beta Functions and Other Gamma-Related Functions 16
1.1.6 Gaussian Hypergeometric Functions 20
1.1.7 Confluent Hypergeometric Functions (Kummer’s Functions) 23
1.1.8 Generalized Hypergeometric Functions 26
1.1.9 Bernoulli and Euler Numbers and Polynomials 29
1.1.10 Integral Transforms 32
1.1.11 Orthogonal Polynomials 32
1.1.12 Basic Hypergeometric Series 34
1.2 Probability and Statistical Preliminaries 37
1.2.1 Calculus of Probabilities 37
1.2.2 Bayes’s Theorem 41
1.2.3 Random Variables 43
1.2.4 Survival Concepts 45
1.2.5 Expected Values 47
1.2.6 Inequalities 49
1.2.7 Moments and Moment Generating Functions 50
1.2.8 Cumulants and Cumulant Generating Functions 54
1.2.9 Joint Moments and Cumulants 56
1.2.10 Characteristic Functions 57
1.2.11 Probability Generating Functions 58
1.2.12 Order Statistics 61
1.2.13 Truncation and Censoring 62
1.2.14 Mixture Distributions 64
1.2.15 Variance of a Function 65
1.2.16 Estimation 66
1.2.17 General Comments on the Computer Generation of Discrete Random Variables 71
1.2.18 Computer Software 73
2 Families of Discrete Distributions 74
2.1 Lattice Distributions 74
2.2 Power Series Distributions 75
2.2.1 Generalized Power Series Distributions 75
2.2.2 Modified Power Series Distributions 79
2.3 Difference-Equation Systems 82
2.3.1 Katz and Extended Katz Families 82
2.3.2 Sundt and Jewell Family 85
2.3.3 Ord’s Family 87
2.4 Kemp Families 89
2.4.1 Generalized Hypergeometric Probability Distributions 89
2.4.2 Generalized Hypergeometric Factorial Moment Distributions 96
2.5 Distributions Based on Lagrangian Expansions 99
2.6 Gould and Abel Distributions 101
2.7 Factorial Series Distributions 103
2.8 Distributions of Order-k 105
2.9 q-Series Distributions 106
3 Binomial Distribution 108
3.1 Definition 108
3.2 Historical Remarks and Genesis 109
3.3 Moments 109
3.4 Properties 112
3.5 Order Statistics 116
3.6 Approximations, Bounds, and Transformations 116
3.6.1 Approximations 116
3.6.2 Bounds 122
3.6.3 Transformations 123
3.7 Computation, Tables, and Computer Generation 124
3.7.1 Computation and Tables 124
3.7.2 Computer Generation 125
3.8 Estimation 126
3.8.1 Model Selection 126
3.8.2 Point Estimation 126
3.8.3 Confidence Intervals 130
3.8.4 Model Verification 133
3.9 Characterizations 134
3.10 Applications 135
3.11 Truncated Binomial Distributions 137
3.12 Other Related Distributions 140
3.12.1 Limiting Forms 140
3.12.2 Sums and Differences of Binomial-Type Variables 140
3.12.3 Poissonian Binomial, Lexian, and Coolidge Schemes 144
3.12.4 Weighted Binomial Distributions 149
3.12.5 Chain Binomial Models 151
3.12.6 Correlated Binomial Variables 151
4 Poisson Distribution 156
4.1 Definition 156
4.2 Historical Remarks and Genesis 156
4.2.1 Genesis 156
4.2.2 Poissonian Approximations 160
4.3 Moments 161
4.4 Properties 163
4.5 Approximations, Bounds, and Transformations 167
4.6 Computation, Tables, and Computer Generation 170
4.6.1 Computation and Tables 170
4.6.2 Computer Generation 171
4.7 Estimation 173
4.7.1 Model Selection 173
4.7.2 Point Estimation 174
4.7.3 Confidence Intervals 176
4.7.4 Model Verification 178
4.8 Characterizations 179
4.9 Applications 186
4.10 Truncated and Misrecorded Poisson Distributions 188
4.10.1 Left Truncation 188
4.10.2 Right Truncation and Double Truncation 191
4.10.3 Misrecorded Poisson Distributions 193
4.11 Poisson–Stopped Sum Distributions 195
4.12 Other Related Distributions 196
4.12.1 Normal Distribution 196
4.12.2 Gamma Distribution 196
4.12.3 Sums and Differences of Poisson Variates 197
4.12.4 Hyper-Poisson Distributions 199
4.12.5 Grouped Poisson Distributions 202
4.12.6 Heine and Euler Distributions 205
4.12.7 Intervened Poisson Distributions 205
5 Negative Binomial Distribution 208
5.1 Definition 208
5.2 Geometric Distribution 210
5.3 Historical Remarks and Genesis of Negative Binomial Distribution 212
5.4 Moments 215
5.5 Properties 217
5.6 Approximations and Transformations 218
5.7 Computation and Tables 220
5.8 Estimation 222
5.8.1 Model Selection 222
5.8.2 P Unknown 222
5.8.3 Both Parameters Unknown 223
5.8.4 Data Sets with a Common Parameter 226
5.8.5 Recent Developments 227
5.9 Characterizations 228
5.9.1 Geometric Distribution 228
5.9.2 Negative Binomial Distribution 231
5.10 Applications 232
5.11 Truncated Negative Binomial Distributions 233
5.12 Related Distributions 236
5.12.1 Limiting Forms 236
5.12.2 Extended Negative Binomial Model 237
5.12.3 Lagrangian Generalized Negative Binomial Distribution 239
5.12.4 Weighted Negative Binomial Distributions 240
5.12.5 Convolutions Involving Negative Binomial Variates 241
5.12.6 Pascal–Poisson Distribution 243
5.12.7 Minimum (Riff–Shuffle) and Maximum Negative Binomial Distributions 244
5.12.8 Condensed Negative Binomial Distributions 246
5.12.9 Other Related Distributions 247
6 Hypergeometric Distributions 251
6.1 Definition 251
6.2 Historical Remarks and Genesis 252
6.2.1 Classical Hypergeometric Distribution 252
6.2.2 Beta–Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution 253
6.2.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution 256
6.2.4 Polya Distributions 258
6.2.5 Hypergeometric Distributions in General 259
6.3 Moments 262
6.4 Properties 265
6.5 Approximations and Bounds 268
6.6 Tables Computation and Computer Generation 271
6.7 Estimation 272
6.7.1 Classical Hypergeometric Distribution 273
6.7.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution 274
6.7.3 Beta–Pascal Distribution 276
6.8 Characterizations 277
6.9 Applications 279
6.9.1 Classical Hypergeometric Distribution 279
6.9.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution 281
6.9.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution 283
6.10 Special Cases 283
6.10.1 Discrete Rectangular Distribution 283
6.10.2 Distribution of Leads in Coin Tossing 286
6.10.3 Yule Distribution 287
6.10.4 Waring Distribution 289
6.10.5 Narayana Distribution 291
6.11 Related Distributions 293
6.11.1 Extended Hypergeometric Distributions 293
6.11.2 Generalized Hypergeometric Probability Distributions 296
6.11.3 Generalized Hypergeometric Factorial Moment Distributions 298
6.11.4 Other Related Distributions 299
7 Logarithmic and Lagrangian Distributions 302
7.1 Logarithmic Distribution 302
7.1.1 Definition 302
7.1.2 Historical Remarks and Genesis 303
7.1.3 Moments 305
7.1.4 Properties 307
7.1.5 Approximations and Bounds 309
7.1.6 Computation, Tables, and Computer Generation 310
7.1.7 Estimation 311
7.1.8 Characterizations 315
7.1.9 Applications 316
7.1.10 Truncated and Modified Logarithmic Distributions 317
7.1.11 Generalizations of the Logarithmic Distribution 319
7.1.12 Other Related Distributions 321
7.2 Lagrangian Distributions 325
7.2.1 Otter’s Multiplicative Process 326
7.2.2 Borel Distribution 328
7.2.3 Consul Distribution 329
7.2.4 Geeta Distribution 330
7.2.5 General Lagrangian Distributions of the First Kind 331
7.2.6 Lagrangian Poisson Distribution 336
7.2.7 Lagrangian Negative Binomial Distribution 340
7.2.8 Lagrangian Logarithmic Distribution 341
7.2.9 Lagrangian Distributions of the Second Kind 342
8 Mixture Distributions 343
8.1 Basic Ideas 343
8.1.1 Introduction 343
8.1.2 Finite Mixtures 344
8.1.3 Varying Parameters 345
8.1.4 Bayesian Interpretation 347
8.2 Finite Mixtures of Discrete Distributions 347
8.2.1 Parameters of Finite Mixtures 347
8.2.2 Parameter Estimation 349
8.2.3 Zero-Modified and Hurdle Distributions 351
8.2.4 Examples of Zero-Modified Distributions 353
8.2.5 Finite Poisson Mixtures 357
8.2.6 Finite Binomial Mixtures 358
8.2.7 Other Finite Mixtures of Discrete Distributions 359
8.3 Continuous and Countable Mixtures of Discrete Distributions 360
8.3.1 Properties of General Mixed Distributions 360
8.3.2 Properties of Mixed Poisson Distributions 362
8.3.3 Examples of Poisson Mixtures 365
8.3.4 Mixtures of Binomial Distributions 373
8.3.5 Examples of Binomial Mixtures 374
8.3.6 Other Continuous and Countable Mixtures of Discrete Distributions 376
8.4 Gamma and Beta Mixing Distributions 378
9 Stopped-Sum Distributions 381
9.1 Generalized and Generalizing Distributions 381
9.2 Damage Processes 386
9.3 Poisson–Stopped Sum (Multiple Poisson) Distributions 388
9.4 Hermite Distribution 394
9.5 Poisson–Binomial Distribution 400
9.6 Neyman Type A Distribution 403
9.6.1 Definition 403
9.6.2 Moment Properties 405
9.6.3 Tables and Approximations 406
9.6.4 Estimation 407
9.6.5 Applications 409
9.7 Polya–Aeppli Distribution 410
9.8 Generalized Polya–Aeppli (Poisson–Negative Binomial) Distribution 414
9.9 Generalizations of Neyman Type A Distribution 416
9.10 Thomas Distribution 421
9.11 Borel–Tanner Distribution: Lagrangian Poisson Distribution 423
9.12 Other Poisson–Stopped Sum (multiple Poisson) Distributions 425
9.13 Other Families of Stopped-Sum Distributions 426
10 Matching, Occupancy, Runs, and q-Series Distributions 430
10.1 Introduction 430
10.2 Probabilities of Combined Events 431
10.3 Matching Distributions 434
10.4 Occupancy Distributions 439
10.4.1 Classical Occupancy and Coupon Collecting 439
10.4.2 Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac Statistics 444
10.4.3 Specified Occupancy and Grassia–Binomial Distributions 446
10.5 Record Value Distributions 448
10.6 Runs Distributions 450
10.6.1 Runs of Like Elements 450
10.6.2 Runs Up and Down 453
10.7 Distributions of Order k 454
10.7.1 Early Work on Success Runs Distributions 454
10.7.2 Geometric Distribution of Order k 456
10.7.3 Negative Binomial Distributions of Order k 458
10.7.4 Poisson and Logarithmic Distributions of Order k 459
10.7.5 Binomial Distributions of Order k 461
10.7.6 Further Distributions of Order k 463
10.8 q-Series Distributions 464
10.8.1 Terminating Distributions 465
10.8.2 q-Series Distributions with Infinite Support 470
10.8.3 Bilateral q-Series Distributions 474
10.8.4 q-Series Related Distributions 476
11 Parametric Regression Models and Miscellanea 478
11.1 Parametric Regression Models 478
11.1.1 Introduction 478
11.1.2 Tweedie–Poisson Family 480
11.1.3 Negative Binomial Regression Models 482
11.1.4 Poisson Lognormal Model 483
11.1.5 Poisson–Inverse Gaussian (Sichel) Model 484
11.1.6 Poisson Polynomial Distribution 487
11.1.7 Weighted Poisson Distributions 488
11.1.8 Double-Poisson and Double-Binomial Distributions 489
11.1.9 Simplex–Binomial Mixture Model 490
11.2 Miscellaneous Discrete Distributions 491
11.2.1 Dandekar’s Modified Binomial and Poisson Models 491
11.2.2 Digamma and Trigamma Distributions 492
11.2.3 Discrete Ades Distribution 494
11.2.4 Discrete Bessel Distribution 495
11.2.5 Discrete Mittag–Leffler Distribution 496
11.2.6 Discrete Student’s t Distribution 498
11.2.7 Feller–Arley and Gegenbauer Distributions 499
11.2.8 Gram–Charlier Type B Distributions 501
11.2.9 “Interrupted” Distributions 502
11.2.10 Lost-Games Distributions 503
11.2.11 Luria–Delbruck Distribution 505
11.2.12 Naor’s Distribution 507
11.2.13 Partial-Sums Distributions 508
11.2.14 Queueing Theory Distributions 512
11.2.15 Reliability and Survival Distributions 514
11.2.16 Skellam–Haldane Gene Frequency Distribution 519
11.2.17 Steyn’s Two-Parameter Power Series Distributions 521
11.2.18 Univariate Multinomial-Type Distributions 522
11.2.19 Urn Models with Stochastic Replacements 524
11.2.20 Zipf-Related Distributions 526
11.2.21 Haight’s Zeta Distributions 533
Bibliography 535
Abbreviations 631
Index 633
