Dirichlet and Related Distributions Theory Methods and Applications 1st Edition by Kai Wang Ng, Guo Liang Tian, Man Lai Tang ISBN 1119998419 978-1119998419

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ISBN 10: 1119998419
ISBN 13: 978-1119998419
Author: Kai Wang Ng, Guo Liang Tian, Man Lai Tang 

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.

Dirichlet and Related Distributions Theory Methods and Applications 1st Table of contents:

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 GDD form.

3.8 Statistical inferences: likelihood function beyond GDD form.

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 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 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 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 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.

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Tags: Kai Wang Ng, Guo Liang Tian, Man Lai Tang, Related Distributions, Theory Methods