Continuous Multivariate Distributions, Volume 1 Models and Applications

by Kotz, Samuel; Balakrishnan, Narayanaswamy; Johnson, Norman L.

• ISBN13: 9780471183877
• ISBN10: 0471183873
• Edition: 2nd
• Format: Hardcover
• Copyright: 2019-01-17
• Publisher: Wiley-Interscience

Summary

Continuous Multivariate Distributions, Volume 1, Second Edition provides a remarkably comprehensive, self-contained resource for this critical statistical area. It covers all significant advances that have occurred in the field over the past quarter century in the theory, methodology, inferential procedures, computational and simulational aspects, and applications of continuous multivariate distributions. In-depth coverage includes MV systems of distributions, MV normal, MV exponential, MV extreme value, MV beta, MV gamma, MV logistic, MV Liouville, and MV Pareto distributions, as well as MV natural exponential families, which have grown immensely since the 1970s. Each distribution is presented in its own chapter along with descriptions of real-world applications gleaned from the current literature on continuous multivariate distributions and their applications.

Author Biography

Norman L. Johnson, PhD, DSc, is Professor Emeritus in the Department of Statistics at the University of North Carolina at Chapel Hill.

Table of Contents

Preface	xv
List of Tables	xix
List of Figures	xxi
Systems of Continuous Multivariate Distributions	1	(104)
Introduction	1	(3)
Historical Remarks	4	(2)
Multivariate Generalization of Pearson System	6	(4)
Series Expansions and Multivariate Central Limit Theorems	10	(10)
Translation Systems	20	(8)
Multivariate Linear Exponential-Type Distributions	28	(2)
Sarmanov's Distributions	30	(3)
Multivariate Linnik's Distributions	33	(1)
Multivariate Kagan's Distributions	34	(2)
Generation of Multivariate Nonnormal Random Variables	36	(8)
Frechet Bounds	44	(3)
Frechet, Plackett and Mardia's Systems	47	(4)
Farlie-Gumbel-Morgenstern Distributions	51	(11)
Multivariate Phase-Type Distributions	62	(1)
Chebyshev-Type and Bonferroni Inequalities	63	(4)
Singular Distributions	67	(2)
Distributions with Almost-Lack of Memory	69	(1)
Distributions with Specified Conditionals	70	(3)
Distributions with Given Marginals	73	(4)
Measures of Multivariate Skewness and Kurtosis	77	(8)
Bibliography	85	(20)
Multivariate Normal Distributions	105	(146)
Introduction and Genesis	105	(2)
Definition and Moments	107	(4)
Other Properties	111	(5)
Order Statistics	116	(5)
Evaluation of Multivariate Normal Probabilities	121	(24)
Reduction Formulas	125	(2)
Orthant Probabilities	127	(6)
Some Special Cases	133	(6)
Approximations	139	(6)
Quadrivariate Normal Orthant Probabilities	145	(6)
Characterizations	151	(10)
Estimation	161	(39)
Estimation of ξ	161	(17)
Estimation of V	178	(4)
Estimation of Correlations	182	(1)
Estimation Under Missing Data	183	(4)
Estimation Under Special Structures	187	(9)
Estimation of Functions of ξ and V	196	(4)
Tolerance Regions	200	(4)
Truncated Multivariate Normal Distributions	204	(7)
Related Distributions	211	(9)
Mixtures of Multivariate Normal Distributions	220	(2)
Complex Multivariate Normal Distributions	222	(29)
Bibliography	223	(28)
Bivariate and Trivariate Normal Distributions	251	(98)
Definition and Applications	251	(1)
Historical Remarks	252	(1)
Properties and Moments	253	(11)
Bivariate Normal Integral-Tables and Approximations	264	(13)
Characterizations	277	(4)
Order Statistics	281	(6)
Trivariate Normal Integral	287	(6)
Estimation	293	(18)
Bivariate Normal Distribution	293	(17)
Trivariate Normal Distributions	310	(1)
Truncated Bivariate and Trivariate Normal Distributions	311	(9)
Dichotomized Variables	320	(6)
Tetrachoic Correlation	320	(4)
Biserial Correlation	324	(2)
Related Distributions	326	(5)
Mixtures of Bivariate Normal Distributions	326	(1)
Bivariate Half Normal Distribution	326	(1)
Distribution of Ratios	327	(1)
``Bivariate Normal'' Distribution with Centered Normal Conditionals	328	(3)
Bivariate Skew-Normal Distribution	331	(18)
Bibliography	333	(16)
Multivariate Exponential Distributions	349	(82)
Introduction	349	(1)
Bivariate Exponential Distributions	350	(37)
Introduction	350	(1)
Gumbel's Bivariate Exponentials	350	(5)
Freund's Bivariate Exponential (Bivariate Exponential Mixture Distributions)	355	(7)
Marshall and Olkin's Bivariate Exponential	362	(7)
Friday and Patil's Bivariate Exponential	369	(1)
Arnold and Strauss' Bivariate Exponential	370	(1)
Moran and Downton's Bivariate Exponential	371	(6)
Singpurwalla and Youngren's Bivariate Exponential	377	(1)
Raftery's Bivariate Exponential	377	(2)
Hayakawa's Bivariate Exponential	379	(1)
Lindley and Singpurwalla's Bivariate Exponential Mixture	380	(3)
Ghurye's Extended Bivariate Lack of Memory Distribution	383	(2)
Cowan's Bivariate Exponential	385	(1)
Infinite Divisibility	385	(1)
Characterizations	386	(1)
Multivariate Exponential Distributions	387	(20)
Freund's Multivariate Exponential	388	(3)
Marshall and Olkin's Multivariate Exponential	391	(5)
Block and Basu's Multivariate Exponential	396	(2)
Olkin and Tong's Multivariate Exponential	398	(1)
Marshall and Olkin's Multivariate Exponential with Limited Memory	399	(1)
Moran and Downton's Multivariate Exponential	400	(1)
Raftery's Multivariate Exponential	401	(2)
Krishnamoorthy and Parthasarathy's Multivariate Exponential	403	(1)
Characterizations	403	(4)
Multivariate Weibull Distributions	407	(5)
Bivariate Distributions Induced by Frailties	412	(19)
Bibliography	416	(15)
Multivariate Gamma Distributions	431	(54)
Introduction	431	(1)
Bivariate Gamma Distributions	432	(22)
McKay's Bivariate Gamma	432	(1)
Cheriyan and Ramabhadran's Bivariate Gamma	432	(4)
Kibble and Moran's Bivariate Gamma	436	(1)
Sarmanov's Bivariate Gamma	437	(1)
Jensen's Bivariate Gamma	438	(2)
Royen's Bivariate Gamma	440	(1)
Farlie--Gumbel--Morgenstern-Type Bivariate Gamma	441	(1)
Prekopa and Szantai's Bivariate Gamma	442	(1)
Smith, Adelfang and Tubb's Bivariate Gamma	443	(2)
Dussauchoy and Berland's Bivariate Gamma	445	(1)
Becker and Roux and Steel and le Roux's Bivariate Gamma	446	(4)
Schmeiser and Lal's Bivariate Gamma	450	(1)
Bivariate Chi-Square Distributions	451	(3)
Multivariate Gamma Distributions	454	(17)
Cheriyan and Ramabhadran's Multivariate Gamma	454	(2)
Gaver's Multivariate Gamma	456	(1)
Krishnamoorthy and Parthasarathy's Multivariate Gamma and Its Extension	457	(1)
Prekopa and Szantai's Multivariate Gamma	458	(1)
Kowalczyk and Tyrcha's Multivariate Gamma	459	(2)
Royen's Multivariate Gamma	461	(4)
Mathai and Moschopoulos's Multivariate Gamma	465	(5)
Dussauchoy and Berland's Multivariate Gamma	470	(1)
Multivariate (Jensen-Type) Chi-Square Distributions	471	(4)
Noncentral Multivariate Chi-Square (Gamma) Distributions	475	(2)
Infinite Divisibility of Multivariate Gamma	477	(8)
Bibliography	478	(7)
Dirichlet and Inverted Dirichlet Distributions	485	(44)
Dirichlet Distributions	485	(6)
Inverted Dirichlet Distribution	491	(4)
Characteristic Functions	495	(2)
Evaluation of Probability Integrals of Dirichlet Distributions	497	(3)
Characterizations	500	(4)
Estimation	504	(3)
Applications	507	(5)
Generalizations	512	(17)
Generalized Dirichlet Distribution as a Prior for Multinomial Parameters	512	(2)
Antelman's Generalization	514	(1)
Johnson and Kotz's Generalization	515	(1)
Delta-Dirichlet Distributions	515	(4)
Connor and Mosimann's Generalization	519	(4)
Bibliography	523	(6)
Multivariate Liouville Distributions	529	(22)
Introduction	529	(1)
Definitions	530	(1)
Properties	531	(5)
Moments and Covariance Structure	536	(2)
Characterizations	538	(3)
Estimation and Applications	541	(3)
Sign-Symmetric Dirichlet and Liouville Distributions	544	(3)
Multivariate p-order Liouville Distributions	547	(4)
Bibliography	548	(3)
Multivariate Logistic Distributions	551	(26)
Introduction	551	(1)
Gumbel-Malik-Abraham Distribution	552	(7)
Frailty and Archimedean Distributions	559	(2)
Farlie--Gumbel--Morgenstern Distributions	561	(1)
Differences of Extreme Value Variables	562	(2)
Mixture Forms	564	(1)
Geometric Minima and Maxima	565	(3)
A General Flexible Model	568	(1)
Conditionally Specified Logistic Model	569	(1)
Some Other Generalizations	570	(7)
Bibliography	573	(4)
Multivariate Pareto Distributions	577	(44)
Introduction	577	(1)
Forms of Univariate Pareto Distributions	577	(2)
Bivariate Pareto Distributions	579	(20)
Bivariate Pareto of the First Kind	579	(5)
Mardia's Bivariate Pareto of the Second Kind	584	(1)
Bivariate Pareto of the Fourth Kind	585	(2)
Conditionally Specified Bivariate Pareto	587	(8)
Muliere and Scarsini's Bivariate Pareto	595	(1)
Bilateral Bivariate Pareto	596	(2)
Bivariate Semi-Pareto	598	(1)
Multivariate Pareto Distributions	599	(22)
Multivariate Pareto of the First Kind	599	(3)
Multivariate Pareto of the Second Kind	602	(3)
Multivariate Pareto of the Third Kind	605	(1)
Multivariate Pareto of the Fourth Kind	606	(5)
Conditionally Specified Multivariate Pareto	611	(1)
Marshall-Olkin-Type Multivariate Pareto	612	(2)
Multivariate Semi-Pareto	614	(1)
Bibliography	615	(6)
Bivariate and Multivariate Extreme Value Distributions	621	(38)
General Bivariate Extreme Value Distributions	622	(2)
Definition	622	(1)
Properties	622	(2)
Special Bivariate Extreme Value Distributions	624	(16)
Type A Distributions	626	(2)
Type B Distributions	628	(1)
Type C Distributions	629	(3)
Normal-Like Bivariate Extreme Value Distributions	632	(5)
Estimation	637	(3)
Multivariate Extreme Value Distributions	640	(19)
Bibliography	652	(7)
Natural Exponential Families	659	(38)
Introduction	659	(1)
Multivariate Natural Exponential Families	660	(5)
Multivariate General Exponential Families	665	(2)
Parameterization by the Mean and Steepness	667	(2)
Variance Function	669	(1)
Affine Transformations and Convolution: The Exponential Dispersion Model	670	(3)
Some Statistical Results for NEFs	673	(5)
Sufficiency, MLE, and UMVUE	673	(2)
Cuts in NEFs	675	(1)
Bayesian Theory	676	(2)
NEFs with Quadratic Variance Function	678	(19)
Morris Class	678	(3)
Multivariate Case	681	(8)
Characterizations	689	(2)
Extensions	691	(1)
Bibliography	691	(6)
Author Index	697	(16)
Subject Index	713