Norman L.Johnson – Univariate Discrete Distributions

Norman L.Johnson – Univariate Discrete Distributions

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Description

This Set Contains: Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Continuous Univariate Distributions, Volume 1, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Continuous Univariate Distributions, Volume 2, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Discrete Multivariate Distributions by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Univariate Discrete Distributions, 3rd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Discover the latest advances in discrete distributions theory The Third Edition of the critically acclaimed Univariate Discrete Distributions provides a self-contained, systematic treatment of the theory, derivation, and application of probability distributions for count data. Generalized zeta-function and q-series distributions have been added and are covered in detail. New families of distributions, including Lagrangian-type distributions, are integrated into this thoroughly revised and updated text. Additional applications of univariate discrete distributions are explored to demonstrate the flexibility of this powerful method. A thorough survey of recent statistical literature draws attention to many new distributions and results for the classical distributions. Approximately 450 new references along with several new sections are introduced to reflect the current literature and knowledge of discrete distributions. Beginning with mathematical, probability, and statistical fundamentals, the authors provide clear coverage of the key topics in the field, including:

• Families of discrete distributions
• Binomial distribution
• Poisson distribution
• Negative binomial distribution
• Hypergeometric distributions
• Logarithmic and Lagrangian distributions
• Mixture distributions
• Stopped-sum distributions
• Matching, occupancy, runs, and q-series distributions
• Parametric regression models and miscellanea

Emphasis continues to be placed on the increasing relevance of Bayesian inference to discrete distribution, especially with regard to the binomial and Poisson distributions. New derivations of discrete distributions via stochastic processes and random walks are introduced without unnecessarily complex discussions of stochastic processes. Throughout the Third Edition, extensive information has been added to reflect the new role of computer-based applications. With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians.  

Table of Contents

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  

Author Information

NORMAN L. JOHNSON, PHD, was Professor Emeritus, Department of Statistics, University of North Carolina at Chapel Hill. Dr. Johnson was Editor-in-Chief (with Dr. Kotz) of the Encyclopedia of Statistical Sciences, Second Edition (Wiley). ADRIENNE W. KEMP, PHD, is Honorary Senior Lecturer at the Mathematical Institute, University of St. Andrews in Scotland. SAMUEL KOTZ, PHD, is Professor and Research Scholar, Department of Engineering Management and Systems Engineering, The George Washington University in Washington, DC.

Reviews

“With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians.”  (Xolosepo, 27 October 2012) “The authors continue to do a praise-worthy job of making the material accessible in the third edition. This book should be on every library’s shelf.” (Journal of the American Statistical Association, September 2006) “These authors have achieved considerable renown for their comprehensive books on statistical distributions.” (Technometrics, August 2006)