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El. knyga: Dirichlet and Related Distributions: Theory, Methods and Applications

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"This book provides a comprehensive review on the Dirichlet distribution including its basic properties, marginal and conditional distributions, cumulative distribution and survival functions"--

"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 familiesof 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"--

"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"--

Extensions of the Dirichlet distribution--particularly grouped and nested versions--have recently begun being used for the statistical analysis of incomplete categorical data, but information about the phenomena is scattered in journal articles. Ng, Guo-Liang Tian (both statistics and actuarial science, U. of Hong Kong) and Man-Lai Tang (mathematics, Hong Kong Baptist U.) present a single volume that systematically reviews the results and their underlying relationships, delineates methods for generating random vectors following these new distributions, and shows some of their important applications in practice. They intend it to serve as a graduate textbook in theoretical and applied statistics and biostatistics. It assumes previous knowledge of standard probability and statistics and basic multivariate analysis. Annotation ©2011 Book News, Inc., Portland, OR (booknews.com)

The Dirichlet distribution appears in many areas of application, which include modelling of compositional data, Bayesian analysis, statistical genetics, and nonparametric inference. This book provides a comprehensive review of the Dirichlet distribution and two extended versions, the Grouped Dirichlet Distribution (GDD) and the Nested Dirichlet Distribution (NDD), arising from likelihood and Bayesian analysis of incomplete categorical data and survey data with non-response.

The theoretical properties and applications are also reviewed in detail for other related distributions, such as the inverted Dirichlet distribution, Dirichlet-multinomial distribution, the truncated Dirichlet distribution, the generalized Dirichlet distribution, Hyper-Dirichlet distribution, scaled Dirichlet distribution, mixed Dirichlet distribution, Liouville distribution, and the generalized Liouville distribution.

Key Features:

  • Presents many of the results and applications that are scattered throughout the literature in one single volume.
  • Looks at the most recent results such as survival function and characteristic function for the uniform distributions over the hyper-plane and simplex; distribution for linear function of Dirichlet components; estimation via the expectation-maximization gradient algorithm and application; etc.
  • Likelihood and Bayesian analyses of incomplete categorical data by using GDD, NDD, and the generalized Dirichlet distribution are illustrated in detail through the EM algorithm and data augmentation structure.
  • Presents a systematic exposition of the Dirichlet-multinomial distribution for multinomial data with extra variation which cannot be handled by the multinomial distribution.
  • S-plus/R codes are featured along with practical examples illustrating the methods.

Practitioners and researchers working in areas such as medical science, biological science and social science will benefit from this book.

Recenzijos

The book is a treasure chest both for researchers in (mathematical and applied) statistics and for practitioners. Researchers will especially pro_t from the impressive survey of the literature and the many references, while practitioners will acknowledge the many real data examples and the S-PLUS code provided in the appendix.  (Zentralblatt MATH, 1 December 2012)

 

Preface xiii
Acknowledgments xv
List of abbreviations xvii
List of symbols xix
List of figures xxiii
List of tables xxv
1 Introduction 1(36)
1.1 Motivating examples
2(5)
1.2 Stochastic representation and the = operator
7(6)
1.2.1 Definition of stochastic representation
7(4)
1.2.2 More properties on the = operator
11(2)
1.3 Beta and inverted beta distributions
13(3)
1.4 Some useful identities and integral formulae
16(1)
1.4.1 Partial-fraction expansion
16(1)
1.4.2 Cambanis–Keener–Simons integral formulae
16(1)
1.4.3 Hermite–Genocchi integral formula
17(1)
1.5 The Newton–Raphson algorithm
17(1)
1.6 Likelihood in missing-data problems
18(5)
1.6.1 Missing-data mechanism
18(1)
1.6.2 The expectation–maximization (EM) algorithm
19(3)
1.6.3 The expectation/conditional maximization (ECM) algorithm
22(1)
1.6.4 The EM gradient algorithm
22(1)
1.7 Bayesian MDPs and inversion of Bayes' formula
23(7)
1.7.1 The data augmentation (DA) algorithm
23(2)
1.7.2 True nature of Bayesian MDP: inversion of Bayes' formula
25(1)
1.7.3 Explicit solution to the DA integral equation
26(3)
1.7.4 Sampling issues in Bayesian MDPs
29(1)
1.8 Basic statistical distributions
30(7)
1.8.1 Discrete distributions
30(2)
1.8.2 Continuous distributions
32(5)
2 Dirichlet distribution 37(60)
2.1 Definition and basic properties
38(5)
2.1.1 Density function and moments
38(2)
2.1.2 Stochastic representations and mode
40(3)
2.2 Marginal and conditional distributions
43(2)
2.3 Survival function and cumulative distribution function
45(6)
2.3.1 Survival function
45(1)
2.3.2 Cumulative distribution function
46(5)
2.4 Characteristic functions
51(6)
2.4.1 The characteristic function of u approximately = U(Tn)
51(2)
2.4.2 The characteristic function of v approximately = U(Vn)
53(2)
2.4.3 The characteristic function of a Dirichlet random vector
55(2)
2.5 Distribution for linear function of a Dirichlet random vector
57(7)
2.5.1 Density for linear function of v approximately = U(Vn)
57(2)
2.5.2 Density for linear function of u approximately = U(Tn)
59(2)
2.5.3 A unified approach to linear functions of variables and order statistics
61(2)
2.5.4 Cumulative distribution function for linear function of a Dirichlet random vector
63(1)
2.6 Characterizations
64(8)
2.6.1 Mosimann's characterization
64(1)
2.6.2 Darroch and Ratcliff's characterization
65(4)
2.6.3 Characterization through neutrality
69(1)
2.6.4 Characterization through complete neutrality
70(2)
2.6.5 Characterization through global and local parameter independence
72(1)
2.7 MLEs of the Dirichlet parameters
72(5)
2.7.1 MLE via the Newton-Raphson algorithm
72(4)
2.7.2 MLE via the EM gradient algorithm
76(1)
2.7.3 Analyzing serum-protein data of Pekin ducklings
76(1)
2.8 Generalized method of moments estimation
77(3)
2.8.1 Method of moments estimation
78(1)
2.8.2 Generalized method of moments estimation
79(1)
2.9 Estimation based on linear models
80(12)
2.9.1 Preliminaries
81(3)
2.9.2 Estimation based on individual linear models
84(3)
2.9.3 Estimation based on the overall linear model
87(5)
2.10 Application in estimating ROC area
92(5)
2.10.1 The ROC curve
92(1)
2.10.2 The ROC area
92(2)
2.10.3 Computing the posterior density of the ROC area
94(1)
2.10.4 Analyzing the mammogram data of breast cancer
95(2)
3 Grouped Dirichlet distribution 97(44)
3.1 Three motivating examples
98(1)
3.2 Density function
99(2)
3.3 Basic properties
101(3)
3.4 Marginal distributions
104(4)
3.5 Conditional distributions
108(2)
3.6 Extension to multiple partitions
110(5)
3.6.1 Density function
110(1)
3.6.2 Some properties
111(1)
3.6.3 Marginal distributions
112(1)
3.6.4 Conditional distributions
113(2)
3.7 Statistical inferences: likelihood function with GDD form
115(6)
3.7.1 Large-sample likelihood inference
116(2)
3.7.2 Small-sample Bayesian inference
118(1)
3.7.3 Analyzing the cervical cancer data
118(1)
3.7.4 Analyzing the leprosy survey data
119(2)
3.8 Statistical inferences: likelihood function beyond GDD form
121(13)
3.8.1 Incomplete 2 x 2 contingency tables: the neurological complication data
121(2)
3.8.2 Incomplete r x c contingency tables
123(9)
3.8.3 Wheeze study in six cities
132(1)
3.8.4 Discussion
133(1)
3.9 Applications under nonignorable missing data mechanism
134(7)
3.9.1 Incomplete r x c tables: nonignorable missing mechanism
134(3)
3.9.2 Analyzing the crime survey data
137(4)
4 Nested Dirichlet distribution 141(34)
4.1 Density function
142(1)
4.2 Two motivating examples
142(2)
4.3 Stochastic representation, mixed moments, and mode
144(4)
4.4 Marginal distributions
148(2)
4.5 Conditional distributions
150(2)
4.6 Connection with exact null distribution for sphericity test
152(1)
4.7 Large-sample likelihood inference
153(6)
4.7.1 Likelihood with NDD form
154(1)
4.7.2 Likelihood beyond NDD form
155(1)
4.7.3 Comparison with existing likelihood strategies
156(3)
4.8 Small-sample Bayesian inference
159(3)
4.8.1 Likelihood with NDD form
159(1)
4.8.2 Likelihood beyond NDD form
159(1)
4.8.3 Comparison with the existing Bayesian strategy
160(2)
4.9 Applications
162(10)
4.9.1 Sample surveys with nonresponse: simulated data
162(1)
4.9.2 Dental caries data
163(3)
4.9.3 Competing-risks model: failure data for radio transmitter receivers
166(3)
4.9.4 Sample surveys: two data sets for death penalty attitude
169(1)
4.9.5 Bayesian analysis of the ultrasound rating data
170(2)
4.10 A brief historical review
172(3)
4.10.1 The neutrality principle
172(2)
4.10.2 The short memory property
174(1)
5 Inverted Dirichlet distribution 175(24)
5.1 Definition through the density function
175(2)
5.1.1 Density function
175(1)
5.1.2 Several useful integral formulae
176(1)
5.1.3 The mixed moment and the mode
177(1)
5.2 Definition through stochastic representation
177(1)
5.3 Marginal and conditional distributions
178(1)
5.4 Cumulative distribution function and survival function
179(4)
5.4.1 Cumulative distribution function
179(3)
5.4.2 Survival function
182(1)
5.5 Characteristic function
183(2)
5.5.1 Univariate case
183(1)
5.5.2 The confluent hypergeometric function of the second kind
183(1)
5.5.3 General case
184(1)
5.6 Distribution for linear function of inverted Dirichlet vector
185(3)
5.6.1 Introduction
185(1)
5.6.2 The distribution of the sum of independent gamma variates
186(1)
5.6.3 The case of two dimensions
187(1)
5.7 Connection with other multivariate distributions
188(4)
5.7.1 Connection with the multivariate t distribution
188(2)
5.7.2 Connection with the multivariate logistic distribution
190(1)
5.7.3 Connection with the multivariate Pareto distribution
191(1)
5.7.4 Connection with the multivariate Cook-Johnson distribution
191(1)
5.8 Applications
192(7)
5.8.1 Bayesian analysis of variance in a linear model
192(3)
5.8.2 Confidence regions for variance ratios in a linear model with random effects
195(4)
6 Dirichlet-multinomial distribution 199(28)
6.1 Probability mass function
199(4)
6.1.1 Motivation
199(1)
6.1.2 Definition via a mixture representation
200(1)
6.1.3 Beta-binomial distribution
201(2)
6.2 Moments of the distribution
203(2)
6.3 Marginal and conditional distributions
205(2)
6.3.1 Marginal distributions
205(1)
6.3.2 Conditional distributions
206(1)
6.3.3 Multiple regression
207(1)
6.4 Conditional sampling method
207(1)
6.5 The method of moments estimation
208(4)
6.5.1 Observations and notations
208(1)
6.5.2 The traditional moments method
209(1)
6.5.3 Mosimann's moments method
210(2)
6.6 The method of maximum likelihood estimation
212(6)
6.6.1 The Newton–Raphson algorithm
212(2)
6.6.2 The Fisher scoring algorithm
214(2)
6.6.3 The EM gradient algorithm
216(2)
6.7 Applications
218(3)
6.7.1 The forest pollen data
218(1)
6.7.2 The teratogenesis data
219(2)
6.8 Testing the multinomial assumption against the Dirichlet–multinomial alternative
221(6)
6.8.1 The likelihood ratio statistic and its null distribution
221(2)
6.8.2 The C(a) test
223(2)
6.8.3 Two illustrative examples
225(2)
7 Truncated Dirichlet distribution 227(20)
7.1 Density function
227(3)
7.1.1 Definition
227(1)
7.1.2 Truncated beta distribution
228(2)
7.2 Motivating examples
230(3)
7.2.1 Case A: matrix A is known
231(1)
7.2.2 Case B: matrix A is unknown
232(1)
7.2.3 Case C: matrix A is partially known
232(1)
7.3 Conditional sampling method
233(4)
7.3.1 Consistent convex polyhedra
233(1)
7.3.2 Marginal distributions
234(1)
7.3.3 Conditional distributions
234(2)
7.3.4 Generation of random vector from a truncated Dirichlet distribution
236(1)
7.4 Gibbs sampling method
237(2)
7.5 The constrained maximum likelihood estimates
239(2)
7.6 Application to misclassification
241(4)
7.6.1 Screening test with binary misclassifications
241(1)
7.6.2 Case–control matched-pair data with polytomous misclassifications
242(3)
7.7 Application to uniform design of experiment with mixtures
245(2)
8 Other related distributions 247(28)
8.1 The generalized Dirichlet distribution
247(7)
8.1.1 Density function
247(3)
8.1.2 Statistical inferences
250(1)
8.1.3 Analyzing the crime survey data
250(2)
8.1.4 Choice of an effective importance density
252(2)
8.2 The hyper-Dirichlet distribution
254(4)
8.2.1 Motivating examples
254(2)
8.2.2 Density function
256(2)
8.3 The scaled Dirichlet distribution
258(5)
8.3.1 Two motivations
258(1)
8.3.2 Stochastic representation and density function
259(1)
8.3.3 Some properties
260(3)
8.4 The mixed Dirichlet distribution
263(6)
8.4.1 Density function
263(1)
8.4.2 Stochastic representation
264(1)
8.4.3 The moments
265(1)
8.4.4 Marginal distributions
266(2)
8.4.5 Conditional distributions
268(1)
8.5 The Liouville distribution
269(3)
8.6 The generalized Liouville distribution
272(3)
Appendix A: Some useful S-plus Codes 275(14)
References 289(14)
Author index 303(4)
Subject index 307
Kai Wang Ng, Department of Statistics and Actuarial Science, The University of Hong Kong. Ng has published over seventy journal articles and book chapters and co-authored five books.

Guo-Liang Tian, Department of Statistics and Actuarial Science, The University, of Hong Kong. His research areas include generalized mixed-effects models for longitudinal data, hierarchical modeling, and applied Bayesian methods in biostatistical models.

Man-Lai Tang, Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
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