Multivariate T-Distributions and Their Applications

by Samuel Kotz , Saralees Nadarajah

• ISBN13: 9780521826549
• ISBN10: 0521826543
• Format: Hardcover
• Copyright: 2004-02-16
• Publisher: Cambridge University Press

Summary

Almost all the results available in the literature on multivariate t-distributions published in the last 50 years are now collected together in this comprehensive reference. Because these distributions are becoming more prominent in many applications, this book is a must for any serious researcher or consultant working in multivariate analysis and statistical distributions. Much of this material has never before appeared in book form. The first part of the book emphasizes theoretical results of a probabilistic nature. In the second part of the book, these are supplemented by a variety of statistical aspects. Various generalizations and applications are dealt with in the final chapters. The material on estimation and regression models is of special value for practitioners in statistics and economics. A comprehensive bibliography of over 350 references is included.

Table of Contents

List of Illustrations	ix
Preface	xi
Introduction	1	(35)
Definition	1	(1)
Representations	2	(5)
Characterizations	7	(1)
A Closure Property	8	(1)
A Consistency Property	9	(1)
Density Expansions	9	(1)
Moments	10	(2)
Maximums	12	(3)
Distribution of a Linear Function	15	(1)
Marginal Distributions	15	(1)
Conditional Distributions	16	(3)
Quadratic Forms	19	(1)
F Matrix	20	(1)
Association	20	(1)
Entropy	21	(2)
Kullback-Leibler Number	23	(3)
Renyi Information	26	(4)
Identities	30	(3)
Some Special Cases	33	(3)
The Characteristic Function	36	(8)
Sutradhar's Approach	36	(2)
Joarder and Ali's Approach	38	(3)
Levy Representation	41	(3)
Linear Combinations, Products, and Ratios	44	(19)
Linear Combinations	44	(8)
Products	52	(4)
Ratios	56	(7)
Bivariate Generalizations and Related Distributions	63	(24)
Owen's Noncentral Bivariate t Distribution	63	(3)
Siddiqui's Noncentral Bivariate t Distribution	66	(2)
Patil and Liao's Noncentral Bivariate t Distribution	68	(1)
Krishnan's Noncentral Bivariate t Distribution	69	(2)
Krishnan's Doubly Noncentral Bivariate t Distribution	71	(1)
Bulgren et al.'s Bivariate t Distribution	72	(1)
Siotani's Noncentral Bivariate t Distribution	73	(1)
Tiku and Kambo's Bivariate t Distribution	74	(2)
Conditionally Specified Bivariate t Distribution	76	(4)
Jones' Bivariate t Distribution	80	(7)
Multivariate Generalizations and Related Distributions	87	(40)
Kshirsagar's Noncentral Multivariate t Distribution	87	(3)
Miller's Noncentral Multivariate t Distribution	90	(1)
Stepwise Multivariate t Distribution	90	(3)
Siotani's Noncentral Multivariate t Distribution	93	(1)
Arellano-Valle and Bolfarine's Generalized t Distribution	94	(3)
Fang et al.'s Asymmetric Multivariate t Distribution	97	(1)
Gupta's Skewed Multivariate t Distribution	98	(4)
Sahu et al.'s Skewed Multivariate t Distribution	102	(1)
Azzalini and Capitanio's Skewed Multivariate t Distribution	103	(2)
Jones' Skewed Multivariate t Distribution	105	(7)
Matrix-Variate t Distribution	112	(7)
Complex Multivariate t Distribution	119	(1)
Steyn's Nonnormal Distributions	120	(6)
Inverted Dirichlet Distribution	126	(1)
Probability Integrals	127	(38)
Dunnett and Sobel's Probability Integrals	127	(4)
Gupta and Sobel's Probability Integrals	131	(4)
John's Probability Integrals	135	(2)
Amos and Bulgren's Probability Integrals	137	(2)
Steffens' Noncentral Probabilities	139	(1)
Dutt's Probability Integrals	140	(3)
Amos' Probability Integral	143	(1)
Fujikoshi's Probability Integrals	144	(1)
Probabilities of Cone	145	(3)
Probabilities of Convex Polyhedra	148	(10)
Probabilities of Linear Inequalities	158	(2)
Maximum Probability Content	160	(1)
Monte Carlo Evaluation	161	(4)
Probability Inequalities	165	(9)
Dunnett and Sobel's Probability Inequalities	165	(4)
Dunn's Probability Inequalities	169	(1)
Halperin's Probability Inequalities	170	(1)
Sidak's Probability Inequalities	171	(1)
Tong's Probability Inequalities	172	(2)
Percentage Points	174	(17)
Dunnett and Sobel's Percentage Points	174	(1)
Krishnaiah and Armitage's Percentage Points	175	(1)
Gupta et al.'s Percentage Points	176	(1)
Rausch and Horn's Percentage Points	176	(1)
Hahn and Hendrickson's Percentage Points	177	(1)
Siotani's Percentage Points	177	(1)
Graybill and Bowden's Percentage Points	178	(2)
Pillai and Ramachandran's Percentage Points	180	(1)
Dunnett's Percentage Points	180	(1)
Gupta and Sobel's Percentage Points	181	(1)
Chen's Percentage Points	182	(1)
Bowden and Graybill's Percentage Points	183	(1)
Dunnett and Tamhane's Percentage Points	183	(4)
Kwong and Liu's Percentage Points	187	(1)
Other Results	187	(4)
Sampling Distributions	191	(16)
Wishart Matrix	191	(7)
Multivariate t Statistic	198	(1)
Hotelling's T2 Statistic	199	(5)
Entropy and Kullback-Leibler Number	204	(3)
Estimation	207	(21)
Tiku and Kambo's Estimation Procedure	207	(3)
ML Estimation via EM Algorithm	210	(2)
Missing Data Imputation	212	(2)
Laplacian T-Approximation	214	(1)
Sutradhar's Score Test	215	(4)
Multivariate t Model	219	(3)
Generalized Multivariate t Model	222	(1)
Simulation	223	(5)
Regression Models	228	(13)
Classical Linear Model	228	(5)
Bayesian Linear Models	233	(2)
Indexed Linear Models	235	(2)
General Linear Model	237	(2)
Nonlinear Models	239	(2)
Applications	241	(6)
Projection Pursuit	241	(2)
Portfolio Optimization	243	(1)
Discriminant and Cluster Analysis	244	(1)
Multiple Decision Problems	245	(1)
Other Applications	246	(1)
References	247	(22)
Index	269