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Dirichlet and Related Distributions: Theory, Methods and Applications

Dirichlet and Related Distributions: Theory, Methods and Applications

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

ISBN: 978-1-119-99841-9

May 2011

336 pages



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.



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.


Author Index.

Subject Index.

“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)


R coding software