Kai Wang Ng & Guo-Liang Tian 
Dirichlet and Related Distributions 
Theory, Methods and Applications

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.

Зміст

Preface.

Acknowledgments.

List of abbreviations.

List of symbols.

List of figures.

List of tables.

1 Introduction.

1.1 Motivating examples.

1.2 Stochastic representation and the¯d=operator.

1.3 Beta and inverted beta distributions.

1.4 Some useful identities and integral formulae.

1.5 The Newton-Raphson algorithm.

1.6 Likelihood in missing-data problems.

1.7 Bayesian MDPs and inversion of bayes’ formula.

1.8 Basic statistical distributions.

2 Dirichlet distribution.

2.1 Definition and basic properties.

2.2 Marginal and conditional distributions.

2.3 Survival function and cumulative distribution function.

2.4 Characteristic functions.

2.5 Distribution for Linear Function of Dirichlet Random Vector.

2.6 Characterizations.

2.7 MLEs of the Dirichlet parameters.

2.8 Generalized method of moments estimation.

2.9 Estimation based on linear models.

2.10 Application in estimating ROC area.

3 Grouped Dirichlet distribution.

3.1 Three motivating examples.

3.2 Density function.

3.3 Basic properties.

3.4 Marginal distributions.

3.5 Conditional distributions.

3.6 Extension to multiple partitions.

3.7 Statistical inferences: likelihood function with GDDform.

3.8 Statistical inferences: likelihood function beyond GDDform.

3.9 Applications under nonignorable missing data mechanism.

4 Nested Dirichlet distribution.

4.1 Density function.

4.2 Two motivating examples.

4.3 Stochastic representation, mixed moments and mode.

4.4 Marginal distributions.

4.5 Conditional distributions.

4.6 Connection with exact null distribution for sphericitytest.

4.7 Large-sample likelihood inference.

4.8 Small-Sample Bayesian inference.

4.9 Applications.

4.10 A brief historical review.

5 Inverted Dirichlet distribution.

5.1 Definition through the density function.

5.2 Definition through stochastic representation.

5.3 Marginal and conditional distributions.

5.4 Cumulative distribution function and survival function.

5.5 Characteristic function.

5.6 Distribution for linear function of inverted Dirichletvector.

5.7 Connection with other multivariate distributions.

5.8 Applications.

6 Dirichlet-multinomial distribution.

6.1 Probability mass function.

6.2 Moments of the distribution.

6.3 Marginal and conditional distributions.

6.4 Conditional sampling method.

6.5 The method of moments estimation.

6.6 The method of maximum likelihood estimation.

6.7 Applications.

6.8 Testing the multinomial assumption against the Dirichlet-multinomial alternative.

7 Truncated Dirichlet distribution.

7.1 Density function.

7.2 Motivating examples.

7.3 Conditional sampling method.

7.4 Gibbs sampling method.

7.5 The constrained maximum likelihood estimates.

7.6 Application to misclassification.

7.7 Application to uniform design of experiment withmixtures.

8 Other related distributions.

8.1 The generalized Dirichlet distribution.

8.2 The hyper-Dirichlet distribution.

8.3 The scaled Dirichlet distribution.

8.4 The mixed Dirichlet distribution.

8.5 The Liouville distribution.

8.6 The generalized Liouville distribution.

Appendix A: Some useful S-plus Codes.

References.

Author Index.

Subject Index.

Про автора

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.