Univariate Discrete Distributions, 3rd Edition

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 Pólya 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 Pólya–Aeppli Distribution, 410

9.8 Generalized Pólya–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 Adès 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–Delbrück 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