Dirichlet and Related Distributions Theory, Methods and Applications

by Ng, Kai Wang; Tian, Guo-Liang; Tang, Man-Lai

• ISBN13: 9780470688199
• ISBN10: 047068819X
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-05-23
• Publisher: Wiley

Summary

This book provides a comprehensive review on the Dirichlet distribution including its basic properties, marginal and conditional distributions, cumulative distribution and survival functions.The authors provide insight into new materials such as survival function, characteristic functions for two uniform distributions over the hyper-plane and simplex distribution for linear function of Dirichlet components estimation via the expectation-maximization gradient algorithm and application. Two new families of distributions (GDD and NDD) are explored, with emphasis on applications in incomplete categorical data and survey data with non-response.Theoretical results on inverted Dirichlet distribution and its applications are featured along with new results that deal with truncated Dirichlet distribution, Dirichlet process and smoothed Dirichlet distribution. The final chapters look at results gathered for Dirichlet-multinomial distribution, Generalized Dirichlet distribution, Liouville distribution, generalized Liouville distribution and matrix-variate Dirichlet distribution.

Author Biography

Kai Wang Ng and Guo-Liang Tian, Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong. Man-Lai Tang, Department of Mathematics, Hong Kong Baptist University, Kowloon Tang, Hong Kong.

Table of Contents

Preface	p. xiii
Acknowledgments	p. xv
List of abbreviations	p. xvii
List of symbols	p. xix
List of figures	p. xxiii
List of tables	p. xxv
Introduction	p. 1
Motivating examples	p. 2
Stochastic representation and the d operator	p. 7
Definition of stochastic representation	p. 7
More properties on the d operator	p. 11
Beta and inverted beta distributions	p. 13
Some useful identities and integral formulae	p. 16
Partial-fraction expansion	p. 16
Cambanis-Keener-Simons integral formulae	p. 16
Hermite-Genocchi integral formula	p. 17
The Newton-Raphson algorithm	p. 17
Likelihood in missing-data problems	p. 18
Missing-data mechanism	p. 18
The expectation-maximization (EM) algorithm	p. 19
The expectation/conditional maximization (ECM) algorithm	p. 22
The EM gradient algorithm	p. 22
Bayesian MDPs and inversion of Bayes' formula	p. 23
The data augmentation (DA) algorithm	p. 23
True nature of Bayesian MDP: inversion of Bayes' formula	p. 25
Explicit solution to the DA integral equation	p. 26
Sampling issues in Bayesian MDPs	p. 29
Basic statistical distributions	p. 30
Discrete distributions	p. 30
Continuous distributions	p. 32
Dirichlet distribution	p. 37
Definition and basic properties	p. 38
Density function and moments	p. 38
Stochastic representations and mode	p. 40
Marginal and conditional distributions	p. 43
Survival function and cumulative distribution function	p. 45
Survival function	p. 45
Cumulative distribution function	p. 46
Characteristic functions	p. 51
The characteristic function of u ∼ U(Tn)	p. 51
The characteristic function of v ∼ U(Vn)	p. 53
The characteristic function of a Dirichlet random vector	p. 55
Distribution for linear function of a Dirichlet random vector	p. 57
Density for linear function of v ∼ U(Vn)	p. 57
Density for linear function of u ∼ U(Tn)	p. 59
A unified approach to linear functions of variables and order statistics	p. 61
Cumulative distribution function for linear function of a Dirichlet random vector	p. 63
Characterizations	p. 64
Mosimann's characterization	p. 64
Darroch and Ratcliff's characterization	p. 65
Characterization through neutrality	p. 69
Characterization through complete neutrality	p. 70
Characterization through global and local parameter independence	p. 72
MLEs of the Dirichlet parameters	p. 72
MLE via the Newton-Raphson algorithm	p. 72
MLE via the EM gradient algorithm	p. 76
Analyzing serum-protein data of Pekin ducklings	p. 76
Generalized method of moments estimation	p. 77
Method of moments estimation	p. 78
Generalized method of moments estimation	p. 79
Estimation based on linear models	p. 80
Preliminaries	p. 81
Estimation based on individual linear models	p. 84
Estimation based on the overall linear model	p. 87
Application in estimating ROC area	p. 92
The ROC curve	p. 92
The ROC area	p. 92
Computing the posterior density of the ROC area	p. 94
Analyzing the mammogram data of breast cancer	p. 95
Grouped Dirichlet distribution	p. 97
Three motivating examples	p. 98
Density function	p. 99
Basic properties	p. 101
Marginal distributions	p. 104
Conditional distributions	p. 108
Extension to multiple partitions	p. 110
Density function	p. 110
Some properties	p. 111
Marginal distributions	p. 112
Conditional distributions	p. 113
Statistical inferences: likelihood function with GDD form	p. 115
Large-sample likelihood inference	p. 116
Small-sample Bayesian inference	p. 118
Analyzing the cervical cancer data	p. 118
Analyzing the leprosy survey data	p. 119
Statistical inferences: likelihood function beyond GDD form	p. 121
Incomplete 2×2 contingency tables: the neurological complication data	p. 121
Incomplete r × c contingency tables	p. 123
Wheeze study in six cities	p. 132
Discussion	p. 133
Applications under nonignorable missing data mechanism	p. 134
Incomplete r × c tables: nonignorable missing mechanism	p. 134
Analyzing the crime survey data	p. 137
Nested Dirichlet distribution	p. 141
Density function	p. 142
Two motivating examples	p. 142
Stochastic representation, mixed moments, and mode	p. 144
Marginal distributions	p. 148
Conditional distributions	p. 150
Connection with exact null distribution for sphericity test	p. 152
Large-sample likelihood inference	p. 153
Likelihood with NDD form	p. 154
Likelihood beyond NDD form	p. 155
Comparison with existing likelihood strategies	p. 156
Small-sample Bayesian inference	p. 159
Likelihood with NDD form	p. 159
Likelihood beyond NDD form	p. 159
Comparison with the existing Bayesian strategy	p. 160
Applications	p. 162
Sample surveys with nonresponse: simulated data	p. 162
Dental caries data	p. 163
Competing-risks model: failure data for radio transmitter receivers	p. 166
Sample surveys: two data sets for death penalty attitude	p. 169
Bayesian analysis of the ultrasound rating data	p. 170
A brief historical review	p. 172
The neutrality principle	p. 172
The short memory property	p. 174
Inverted Dirichlet distribution	p. 175
Definition through the density function	p. 175
Density function	p. 175
Several useful integral formulae	p. 176
The mixed moment and the mode	p. 177
Definition through stochastic representation	p. 177
Marginal and conditional distributions	p. 178
Cumulative distribution function and survival function	p. 179
Cumulative distribution function	p. 179
Survival function	p. 182
Characteristic function	p. 183
Univariate case	p. 183
The confluent hypergeometric function of the second kind	p. 183
General case	p. 184
Distribution for linear function of inverted Dirichlet vector	p. 185
Introduction	p. 185
The distribution of the sum of independent gamma variates	p. 186
The case of two dimensions	p. 187
Connection with other multivariate distributions	p. 188
Connection with the multivariate t distribution	p. 188
Connection with the multivariate logistic distribution	p. 190
Connection with the multivariate Pareto distribution	p. 191
Connection with the multivariate Cook-Johnson distribution	p. 191
Applications	p. 192
Bayesian analysis of variance in a linear model	p. 192
Confidence regions for variance ratios in a linear model with random effects	p. 195
Dirichlet-multinomial distribution	p. 199
Probability mass function	p. 199
Motivation	p. 199
Definition via a mixture representation	p. 200
Beta-binomial distribution	p. 201
Moments of the distribution	p. 203
Marginal and conditional distributions	p. 205
Marginal distributions	p. 205
Conditional distributions	p. 206
Multiple regression	p. 207
Conditional sampling method	p. 207
The method of moments estimation	p. 208
Observations and notations	p. 208
The traditional moments method	p. 209
Mosimann's moments method	p. 210
The method of maximum likelihood estimation	p. 212
The Newton-Raphson algorithm	p. 212
The Fisher scoring algorithm	p. 214
The EM gradient algorithm	p. 216
Applications	p. 218
The forest pollen data	p. 218
The teratogenesis data	p. 219
Testing the multinomial assumption against the Dirichlet-multinomial alternative	p. 221
The likelihood ratio statistic and its null distribution	p. 221
The C(¿) test	p. 223
Two illustrative examples	p. 225
Truncated Dirichlet distribution	p. 227
Density function	p. 227
Definition	p. 227
Truncated beta distribution	p. 228
Motivating examples	p. 230
Case A: matrix ¿ is known	p. 231
Case B: matrix ¿ is unknown	p. 232
Case C: matrix ¿ is partially known	p. 232
Conditional sampling method	p. 233
Consistent convex polyhedra	p. 233
Marginal distributions	p. 234
Conditional distributions	p. 234
Generation of random vector from a truncated Dirichlet distribution	p. 236
Gibbs sampling method	p. 237
The constrained maximum likelihood estimates	p. 239
Application to misclassification	p. 241
Screening test with binary misclassifications	p. 241
Case-control matched-pair data with polytomous misclassifications	p. 242
Application to uniform design of experiment with mixtures	p. 245
Other related distributions	p. 247
The generalized Dirichlet distribution	p. 247
Density function	p. 247
Statistical inferences	p. 250
Analyzing the crime survey data	p. 250
Choice of an effective importance density	p. 252
The hyper-Dirichlet distribution	p. 254
Motivating examples	p. 254
Density function	p. 256
The scaled Dirichlet distribution	p. 258
Two motivations	p. 258
Stochastic representation and density function	p. 259
Some properties	p. 260
The mixed Dirichlet distribution	p. 263
Density function	p. 263
Stochastic representation	p. 264
The moments	p. 265
Marginal distributions	p. 266
Conditional distributions	p. 268
The Liouville distribution	p. 269
The generalized Liouville distribution	p. 272
Some useful S-plus Codes	p. 275
References	p. 289
Author index	p. 303
Subject index	p. 307
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