Univariate Discrete Distributions

by Johnson, Norman L.; Kemp, Adrienne W.; Kotz, Samuel

• ISBN13: 9780471272465
• ISBN10: 0471272469
• Edition: 3rd
• Format: Hardcover
• Copyright: 2005-08-30
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

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). <p>ADRIENNE W. KEMP, PHD, is Honorary Senior Lecturer at the Mathematical Institute, University of St. Andrews in Scotland. <p>SAMUEL KOTZ, PHD, is Professor and Research Scholar, Department of Engineering Management and Systems Engineering, The George Washington University in Washington, DC.

Table of Contents

Preface	xvii
1 Preliminary Information	1	(73)
1.1 Mathematical Preliminaries,	1	(36)
1.1.1 Factorial and Combinatorial Conventions,	1	(4)
1.1.2 Gamma and Beta Functions,	5	(5)
1.1.3 Finite Difference Calculus,	10	(4)
1.1.4 Differential Calculus,	14	(2)
1.1.5 Incomplete Gamma and Beta Functions and Other Gamma-Related Functions,	16	(4)
1.1.6 Gaussian Hypergeometric Functions,	20	(3)
1.1.7 Confluent Hypergeometric Functions (Kummer's Functions),	23	(3)
1.1.8 Generalized Hypergeometric Functions,	26	(3)
1.1.9 Bernoulli and Euler Numbers and Polynomials,	29	(3)
1.1.10 Integral Transforms,	32	(1)
1.1.11 Orthogonal Polynomials,	32	(2)
1.1.12 Basic Hypergeometric Series,	34	(3)
1.2 Probability and Statistical Preliminaries,	37	(37)
1.2.1 Calculus of Probabilities,	37	(4)
1.2.2 Bayes's Theorem,	41	(2)
1.2.3 Random Variables,	43	(2)
1.2.4 Survival Concepts,	45	(2)
1.2.5 Expected Values,	47	(2)
1.2.6 Inequalities,	49	(1)
1.2.7 Moments and Moment Generating Functions,	50	(4)
1.2.8 Cumulants and Cumulant Generating Functions,	54	(2)
1.2.9 Joint Moments and Cumulants,	56	(1)
1.2.10 Characteristic Functions,	57	(1)
1.2.11 Probability Generating Functions,	58	(3)
1.2.12 Order Statistics,	61	(1)
1.2.13 Truncation and Censoring,	62	(2)
1.2.14 Mixture Distributions,	64	(1)
1.2.15 Variance of a Function,	65	(1)
1.2.16 Estimation,	66	(5)
1.2.17 General Comments on the Computer Generation of Discrete Random Variables,	71	(2)
1.2.18 Computer Software,	73	(1)
2 Families of Discrete Distributions	74	(34)
2.1 Lattice Distributions,	74	(1)
2.2 Power Series Distributions,	75	(7)
2.2.1 Generalized Power Series Distributions,	75	(4)
2.2.2 Modified Power Series Distributions,	79	(3)
2.3 Difference-Equation Systems,	82	(7)
2.3.1 Katz and Extended Katz Families,	82	(3)
2.3.2 Sundt and Jewell Family,	85	(2)
2.3.3 Ord's Family,	87	(2)
2.4 Kemp Families,	89	(10)
2.4.1 Generalized Hypergeometric Probability Distributions,	89	(7)
2.4.2 Generalized Hypergeometric Factorial Moment Distributions,	96	(3)
2.5 Distributions Based on Lagrangian Expansions,	99	(2)
2.6 Gould and Abel Distributions,	101	(2)
2.7 Factorial Series Distributions,	103	(2)
2.8 Distributions of Order-kappa,	105	(1)
2.9 q-Series Distributions,	106	(2)
3 Binomial Distribution	108	(48)
3.1 Definition,	108	(1)
3.2 Historical Remarks and Genesis,	109	(1)
3.3 Moments,	109	(3)
3.4 Properties,	112	(4)
3.5 Order Statistics,	116	(1)
3.6 Approximations, Bounds, and Transformations,	116	(8)
3.6.1 Approximations,	116	(6)
3.6.2 Bounds,	122	(1)
3.6.3 Transformations,	123	(1)
3.7 Computation, Tables, and Computer Generation,	124	(2)
3.7.1 Computation and Tables,	124	(1)
3.7.2 Computer Generation,	125	(1)
3.8 Estimation,	126	(8)
3.8.1 Model Selection,	126	(1)
3.8.2 Point Estimation,	126	(4)
3.8.3 Confidence Intervals,	130	(3)
3.8.4 Model Verification,	133	(1)
3.9 Characterizations,	134	(1)
3.10 Applications,	135	(2)
3.11 Truncated Binomial Distributions,	137	(3)
3.12 Other Related Distributions,	140	(16)
3.12.1 Limiting Forms,	140	(1)
3.12.2 Sums and Differences of Binomial-Type Variables,	140	(4)
3.12.3 Poissonian Binomial, Lexian, and Coolidge Schemes,	144	(5)
3.12.4 Weighted Binomial Distributions,	149	(2)
3.12.5 Chain Binomial Models,	151	(1)
3.12.6 Correlated Binomial Variables,	151	(5)
4 Poisson Distribution	156	(52)
4.1 Definition,	156	(1)
4.2 Historical Remarks and Genesis,	156	(5)
4.2.1 Genesis,	156	(4)
4.2.2 Poissonian Approximations,	160	(1)
4.3 Moments,	161	(2)
4.4 Properties,	163	(4)
4.5 Approximations, Bounds, and Transformations,	167	(3)
4.6 Computation, Tables, and Computer Generation,	170	(3)
4.6.1 Computation and Tables,	170	(1)
4.6.2 Computer Generation,	171	(2)
4.7 Estimation,	173	(6)
4.7.1 Model Selection,	173	(1)
4.7.2 Point Estimation,	174	(2)
4.7.3 Confidence Intervals,	176	(2)
4.7.4 Model Verification,	178	(1)
4.8 Characterizations,	179	(7)
4.9 Applications,	186	(2)
4.10 Truncated and Misrecorded Poisson Distributions,	188	(7)
4.10.1 Left Truncation,	188	(3)
4.10.2 Right Truncation and Double Truncation,	191	(2)
4.10.3 Misrecorded Poisson Distributions,	193	(2)
4.11 Poisson–Stopped Sum Distributions,	195	(1)
4.12 Other Related Distributions,	196	(12)
4.12.1 Normal Distribution,	196	(1)
4.12.2 Gamma Distribution,	196	(1)
4.12.3 Sums and Differences of Poisson Variates,	197	(2)
4.12.4 Hyper-Poisson Distributions,	199	(3)
4.12.5 Grouped Poisson Distributions,	202	(3)
4.12.6 Heine and Euler Distributions,	205	(1)
4.12.7 Intervened Poisson Distributions,	205	(3)
5 Negative Binomial Distribution	208	(43)
5.1 Definition,	208	(2)
5.2 Geometric Distribution,	210	(2)
5.3 Historical Remarks and Genesis of Negative Binomial Distribution,	212	(3)
5.4 Moments,	215	(2)
5.5 Properties,	217	(1)
5.6 Approximations and Transformations,	218	(2)
5.7 Computation and Tables,	220	(2)
5.8 Estimation,	222	(6)
5.8.1 Model Selection,	222	(1)
5.8.2 P Unknown,	222	(1)
5.8.3 Both Parameters Unknown,	223	(3)
5.8.4 Data Sets with a Common Parameter,	226	(1)
5.8.5 Recent Developments,	227	(1)
5.9 Characterizations,	228	(4)
5.9.1 Geometric Distribution,	228	(3)
5.9.2 Negative Binomial Distribution,	231	(1)
5.10 Applications,	232	(1)
5.11 Truncated Negative Binomial Distributions,	233	(3)
5.12 Related Distributions,	236	(15)
5.12.1 Limiting Forms,	236	(1)
5.12.2 Extended Negative Binomial Model,	237	(2)
5.12.3 Lagrangian Generalized Negative Binomial Distribution,	239	(1)
5.12.4 Weighted Negative Binomial Distributions,	240	(1)
5.12.5 Convolutions Involving Negative Binomial Variates,	241	(2)
5.12.6 Pascal–Poisson Distribution,	243	(1)
5.12.7 Minimum (Riff–Shuffle) and Maximum Negative Binomial Distributions,	244	(2)
5.12.8 Condensed Negative Binomial Distributions,	246	(1)
5.12.9 Other Related Distributions,	247	(4)
6 Hypergeometric Distributions	251	(51)
6.1 Definition,	251	(1)
6.2 Historical Remarks and Genesis,	252	(10)
6.2.1 Classical Hypergeometric Distribution,	252	(1)
6.2.2 Beta–Binomial Distribution, Negative (Inverse) Hypergeometric Distribution: Hypergeometric Waiting-Time Distribution,	253	(3)
6.2.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution,	256	(2)
6.2.4 Pólya Distributions,	258	(1)
6.2.5 Hypergeometric Distributions in General,	259	(3)
6.3 Moments,	262	(3)
6.4 Properties,	265	(3)
6.5 Approximations and Bounds,	268	(3)
6.6 Tables, Computation, and Computer Generation,	271	(1)
6.7 Estimation,	272	(5)
6.7.1 Classical Hypergeometric Distribution,	273	(1)
6.7.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution,	274	(2)
6.7.3 Beta–Pascal Distribution,	276	(1)
6.8 Characterizations,	277	(2)
6.9 Applications,	279	(4)
6.9.1 Classical Hypergeometric Distribution,	279	(2)
6.9.2 Negative (Inverse) Hypergeometric Distribution: Beta–Binomial Distribution,	281	(2)
6.9.3 Beta–Negative Binomial Distribution: Beta–Pascal Distribution, Generalized Waring Distribution,	283	(1)
6.10 Special Cases,	283	(10)
6.10.1 Discrete Rectangular Distribution,	283	(3)
6.10.2 Distribution of Leads in Coin Tossing,	286	(1)
6.10.3 Yule Distribution,	287	(2)
6.10.4 Waring Distribution,	289	(2)
6.10.5 Narayana Distribution,	291	(2)
6.11 Related Distributions,	293	(9)
6.11.1 Extended Hypergeometric Distributions,	293	(3)
6.11.2 Generalized Hypergeometric Probability Distributions,	296	(2)
6.11.3 Generalized Hypergeometric Factorial Moment Distributions,	298	(1)
6.11.4 Other Related Distributions,	299	(3)
7 Logarithmic and Lagrangian Distributions	302	(41)
7.1 Logarithmic Distribution,	302	(23)
7.1.1 Definition,	302	(1)
7.1.2 Historical Remarks and Genesis,	303	(2)
7.1.3 Moments,	305	(2)
7.1.4 Properties,	307	(2)
7.1.5 Approximations and Bounds,	309	(1)
7.1.6 Computation, Tables, and Computer Generation,	310	(1)
7.1.7 Estimation,	311	(4)
7.1.8 Characterizations,	315	(1)
7.1.9 Applications,	316	(1)
7.1.10 Truncated and Modified Logarithmic Distributions,	317	(2)
7.1.11 Generalizations of the Logarithmic Distribution,	319	(2)
7.1.12 Other Related Distributions,	321	(4)
7.2 Lagrangian Distributions,	325	(18)
7.2.1 Otter's Multiplicative Process,	326	(2)
7.2.2 Borel Distribution,	328	(1)
7.2.3 Consul Distribution,	329	(1)
7.2.4 Geeta Distribution,	330	(1)
7.2.5 General Lagrangian Distributions of the First Kind,	331	(5)
7.2.6 Lagrangian Poisson Distribution,	336	(4)
7.2.7 Lagrangian Negative Binomial Distribution,	340	(1)
7.2.8 Lagrangian Logarithmic Distribution,	341	(1)
7.2.9 Lagrangian Distributions of the Second Kind,	342	(1)
8 Mixture Distributions	343	(38)
8.1 Basic Ideas,	343	(4)
8.1.1 Introduction,	343	(1)
8.1.2 Finite Mixtures,	344	(1)
8.1.3 Varying Parameters,	345	(2)
8.1.4 Bayesian Interpretation,	347	(1)
8.2 Finite Mixtures of Discrete Distributions,	347	(13)
8.2.1 Parameters of Finite Mixtures,	347	(2)
8.2.2 Parameter Estimation,	349	(2)
8.2.3 Zero-Modified and Hurdle Distributions,	351	(2)
8.2.4 Examples of Zero-Modified Distributions,	353	(4)
8.2.5 Finite Poisson Mixtures,	357	(1)
8.2.6 Finite Binomial Mixtures,	358	(1)
8.2.7 Other Finite Mixtures of Discrete Distributions,	359	(1)
8.3 Continuous and Countable Mixtures of Discrete Distributions,	360	(18)
8.3.1 Properties of General Mixed Distributions,	360	(2)
8.3.2 Properties of Mixed Poisson Distributions,	362	(3)
8.3.3 Examples of Poisson Mixtures,	365	(8)
8.3.4 Mixtures of Binomial Distributions,	373	(1)
8.3.5 Examples of Binomial Mixtures,	374	(2)
8.3.6 Other Continuous and Countable Mixtures of Discrete Distributions,	376	(2)
8.4 Gamma and Beta Mixing Distributions,	378	(3)
9 Stopped-Sum Distributions	381	(49)
9.1 Generalized and Generalizing Distributions,	381	(5)
9.2 Damage Processes,	386	(2)
9.3 Poisson–Stopped Sum (Multiple Poisson) Distributions,	388	(6)
9.4 Hermite Distribution,	394	(6)
9.5 Poisson–Binomial Distribution,	400	(3)
9.6 Neyman Type A Distribution,	403	(7)
9.6.1 Definition,	403	(2)
9.6.2 Moment Properties,	405	(1)
9.6.3 Tables and Approximations,	406	(1)
9.6.4 Estimation,	407	(2)
9.6.5 Applications,	409	(1)
9.7 Pólya–Aeppli Distribution,	410	(4)
9.8 Generalized Pólya–Aeppli (Poisson–Negative Binomial) Distribution,	414	(2)
9.9 Generalizations of Neyman Type A Distribution,	416	(5)
9.10 Thomas Distribution,	421	(2)
9.11 Borel–Tanner Distribution: Lagrangian Poisson Distribution,	423	(2)
9.12 Other Poisson–Stopped Sum (multiple Poisson) Distributions,	425	(1)
9.13 Other Families of Stopped-Sum Distributions,	426	(4)
10 Matching, Occupancy, Runs, and q-Series Distributions	430	(48)
10.1 Introduction,	430	(1)
10.2 Probabilities of Combined Events,	431	(3)
10.3 Matching Distributions,	434	(5)
10.4 Occupancy Distributions,	439	(9)
10.4.1 Classical Occupancy and Coupon Collecting,	439	(5)
10.4.2 Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac Statistics,	444	(2)
10.4.3 Specified Occupancy and Grascia–Binomial Distributions,	446	(2)
10.5 Record Value Distributions,	448	(2)
10.6 Runs Distributions,	450	(4)
10.6.1 Runs of Like Elements,	450	(3)
10.6.2 Runs Up and Down,	453	(1)
10.7 Distributions of Order kappa,	454	(10)
10.7.1 Early Work on Success Runs Distributions,	454	(2)
10.7.2 Geometric Distribution of Order kappa,	456	(2)
10.7.3 Negative Binomial Distributions of Order kappa,	458	(1)
10.7.4 Poisson and Logarithmic Distributions of Order kappa,	459	(2)
10.7.5 Binomial Distributions of Order kappa,	461	(2)
10.7.6 Further Distributions of Order kappa,	463	(1)
10.8 q-Series Distributions,	464	(14)
10.8.1 Terminating Distributions,	465	(5)
10.8.2 q-Series Distributions with Infinite Support,	470	(4)
10.8.3 Bilateral q-Series Distributions,	474	(2)
10.8.4 q-Series Related Distributions,	476	(2)
11 Parametric Regression Models and Miscellanea	478	(57)
11.1 Parametric Regression Models,	478	(13)
11.1.1 Introduction,	478	(2)
11.1.2 Tweedle—Poisson Family,	480	(2)
11.1.3 Negative Binomial Regression Models,	482	(1)
11.1.4 Poisson Lognormal Model,	483	(1)
11.1.5 Poisson—Inverse Gaussian (Sichel) Model,	484	(3)
11.1.6 Poisson Polynomial Distribution,	487	(1)
11.1.7 Weighted Poisson Distributions,	488	(1)
11.1.8 Double-Poisson and Double-Binomial Distributions,	489	(1)
11.1.9 Simplex—Bínomíal Mixture Model,	490	(1)
11.2 Miscellaneous Discrete Distributions,	491	(44)
11.2.1 Dandekar's Modified Binomial and Poisson Models,	491	(1)
11.2.2 Digamma and Trigamma Distributions,	492	(2)
11.2.3 Discrete Adès Distribution,	494	(1)
11.2.4 Discrete Bessel Distribution,	495	(1)
11.2.5 Discrete Mittag—Leffler Distribution,	496	(2)
11.2.6 Discrete Student's t Distribution,	498	(1)
11.2.7 Feller—Arley and Gegenbauer Distributions,	499	(2)
11.2.8 Gram—Charlier Type B Distributions,	501	(1)
11.2.9 "Interrupted" Distributions,	502	(1)
11.2.10 Lost-Games Distributions,	503	(2)
11.2.11 Luria—Delbrück Distribution,	505	(2)
11.2.12 Naor's Distribution,	507	(1)
11.2.13 Partial-Sums Distributions,	508	(4)
11.2.14 Queueing Theory Distributions,	512	(2)
11.2.15 Reliability and Survival Distributions,	514	(5)
11.2.16 Skellam—Haldane Gene Frequency Distribution,	519	(2)
11.2.17 Stegn's Two-Parameter Power Series Distributions,	521	(1)
11.2.18 Univariate Multinomial-Type Distributions,	522	(2)
11.2.19 Urn Models with Stochastic Replacements,	524	(2)
11.2.20 Zipf-Related Distributions,	526	(7)
11.2.21 Haight's Zeta Distributions,	533	(2)
Bibliography	535	(96)
Abbreviations	631	(2)
Index	633