Multivariate t distributions and their applications

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
List of Illustrations......Page 10
Preface......Page 12
1.1 Definition......Page 14
1.2 Representations......Page 15
1.3 Characterizations......Page 20
1.4 A Closure Property......Page 21
1.6 Density Expansions......Page 22
1.7 Moments......Page 23
1.8 Maximums......Page 25
1.10 Marginal Distributions......Page 28
1.11 Conditional Distributions......Page 29
1.12 Quadratic Forms......Page 32
1.14 Association......Page 33
1.15 Entropy......Page 34
1.16 Kullback-Leibler Number......Page 36
1.17 Renyi Information......Page 39
1.18 Identities......Page 43
1.19 Some Special Cases......Page 46
2.1 Sutradhar\'s Approach......Page 49
2.2 Joarder and Ali\'s Approach......Page 51
2.3 Levy Representation......Page 54
3.1 Linear Combinations......Page 57
3.2 Products......Page 65
3.3 Ratios......Page 69
4.1 Owen\'s Noncentral Bivariate t Distribution......Page 76
4.2 Siddiqui\'s Noncentral Bivariate t Distribution......Page 79
4.3 Patil and Liao\'s Noncentral Bivariate t Distribution......Page 81
4.4 Krishnan\'s Noncentral Bivariate t Distribution......Page 82
4.5 Krishnan\'s Doubly Noncentral Bivariate t Distribution......Page 84
4.6 Bulgren et al.\'s Bivariate t Distribution......Page 85
4.7 Siotani\'s Noncentral Bivariate t Distribution......Page 86
4.8 Tiku and Kambo\'s Bivariate t Distribution......Page 87
4.9 Conditionally Specified Bivariate t Distribution......Page 89
4.10 Jones\' Bivariate t Distribution......Page 93
5.1 Kshirsagar\'s Noncentral Multivariate t Distribution......Page 100
5.3 Stepwise Multivariate t Distribution......Page 103
5.4 Siotani\'s Noncentral Multivariate t Distribution......Page 106
5.5 Arellano-Valle and Bolfarine\'s Generalized t Distribution......Page 107
5.6 Fang et al.\'s Asymmetric Multivariate t Distribution......Page 110
5.7 Gupta\'s Skewed Multivariate t Distribution......Page 111
5.8 Sahu et al.\'s Skewed Multivariate t Distribution......Page 115
5.9 Azzalini and Capitanio\'s Skewed Multivariate t Distribution......Page 116
5.10 Jones\' Skewed Multivariate t Distribution......Page 118
5.11 Matrix-Variate t Distribution......Page 125
5.12 Complex Multivariate t Distribution......Page 132
5.13 Steyn\'s Nonnormal Distributions......Page 133
5.14 Inverted Dirichlet Distribution......Page 139
6.1 Dunnett and Sobel\'s Probability Integrals......Page 140
6.2 Gupta and Sobel\'s Probability Integrals......Page 144
6.3 John\'s Probability Integrals......Page 148
6.4 Amos and Bulgren\'s Probability Integrals......Page 150
6.5 Steffens\' Noncentral Probabilities......Page 152
6.6 Dutt\'s Probability Integrals......Page 153
6.7 Amos\' Probability Integral......Page 156
6.8 Fujikoshi\'s Probability Integrals......Page 157
6.9 Probabilities of Cone......Page 158
6.10 Probabilities of Convex Polyhedra......Page 161
6.11 Probabilities of Linear Inequalities......Page 171
6.12 Maximum Probability Content......Page 173
6.13 Monte Carlo Evaluation......Page 174
7.1 Dunnett and Sobel\'s Probability Inequalities......Page 178
7.2 Dunn\'s Probability Inequalities......Page 182
7.3 Halperin\'s Probability Inequalities......Page 183
7.4 Sidak\'s Probability Inequalities......Page 184
7.5 Tong\'s Probability Inequalities......Page 185
8.1 Dunnett and Sobel\'s Percentage Points......Page 187
8.2 Krishnaiah and Armitage\'s Percentage Points......Page 188
8.4 Rausch and Horn\'s Percentage Points......Page 189
8.6 Siotani\'s Percentage Points......Page 190
8.7 Graybill and Bowden\'s Percentage Points......Page 191
8.9 Dunnett\'s Percentage Points......Page 193
8.10 Gupta and Sobel\'s Percentage Points......Page 194
8.11 Chen\'s Percentage Points......Page 195
8.13 Dunnett and Tamhane\'s Percentage Points......Page 196
8.15 Other Results......Page 200
9.1 Wishart Matrix......Page 204
9.2 Multivariate t Statistic......Page 211
9.3 Hotelling\'s T2 Statistic......Page 212
9.4 Entropy and Kullback-Leibler Number......Page 217
10.1 Tiku and Kambo\'s Estimation Procedure......Page 220
10.2 ML Estimation via EM Algorithm......Page 223
10.3 Missing Data Imputation......Page 225
10.4 Laplacian T-Approximation......Page 227
10.5 Sutradhar\'s Score Test......Page 228
10.6 Multivariate t Model......Page 232
10.7 Generalized Multivariate t Model......Page 235
10.8 Simulation......Page 236
11.1 Classical Linear Model......Page 241
11.2 Bayesian Linear Models......Page 246
11.3 Indexed Linear Models......Page 248
11.4 General Linear Model......Page 250
11.5 Nonlinear Models......Page 252
12.1 Projection Pursuit......Page 254
12.2 Portfolio Optimization......Page 256
12.3 Discriminant and Cluster Analysis......Page 257
12.4 Multiple Decision Problems......Page 258
12.5 Other Applications......Page 259
References......Page 260
Index......Page 282