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.

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.

Preface.

Acknowledgments.

List of Figures.

List of Tables.

List of Abbreviations.

List of Symbols.

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 Missing Data Problems (MDP) 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 Maximum Likelihood Estimates (MLEs) of the Dirichlet Parameters.

2.8 Generalized Method of Moments Estimation.

2.9 Estimation Based on Linear Models.

2.10 Application in Estimating Receiver Operating Characteristic (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 GDD Form.

3.8 Statistical Inferences: Likelihood Function beyond GDD Form.

3.9 Applications under Non-ignorable 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 Sphericity Test.

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 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 Dirichlet Vector.

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 with Mixtures.

8 Other Related Distributions.

8.1 The Generalized Dirichlet Distribution.

8.2 The Hyperdirichlet Distribution.

8.3 The Scaled Dirichlet Distribution.

8.4 The Mixed Dirichlet Distribution.

8.5 The Liouville Distribution.

8.6 The Generalized Liouville Distribution.

A Some Useful S-plus Codes.

A.1 Multinomial Distribution.

A.2 Dirichlet Distribution.

A.3 Grouped Dirichlet Distribution.

A.4 Nested Dirichlet Distribution.

A.5 Dirichlet-Multinomial Distribution.

References.

Author Index.

Subject Index.