Ng / Tian / Tang Dirichlet and Related Distributions

Theory, Methods and Applications
1. Auflage 2011
ISBN: 978-1-119-99586-9
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 336 Seiten, E-Book

Reihe: Wiley Series in Probability and Statistics

The Dirichlet distribution appears in many areas of application,which include modelling of compositional data, Bayesian analysis,statistical genetics, and nonparametric inference. This bookprovides a comprehensive review of the Dirichlet distribution andtwo extended versions, the Grouped Dirichlet Distribution (GDD) andthe Nested Dirichlet Distribution (NDD), arising from likelihoodand Bayesian analysis of incomplete categorical data and surveydata with non-response.
The theoretical properties and applications are also reviewed indetail for other related distributions, such as the invertedDirichlet distribution, Dirichlet-multinomial distribution, thetruncated Dirichlet distribution, the generalized Dirichletdistribution, Hyper-Dirichlet distribution, scaled Dirichletdistribution, mixed Dirichlet distribution, Liouville distribution,and the generalized Liouville distribution.
Key Features:
* Presents many of the results and applications that arescattered throughout the literature in one single volume.
* Looks at the most recent results such as survival function andcharacteristic function for the uniform distributions over thehyper-plane and simplex; distribution for linear function ofDirichlet components; estimation via the expectation-maximizationgradient algorithm and application; etc.
* Likelihood and Bayesian analyses of incomplete categoricaldata by using GDD, NDD, and the generalized Dirichlet distributionare illustrated in detail through the EM algorithm and dataaugmentation structure.
* Presents a systematic exposition of the Dirichlet-multinomialdistribution for multinomial data with extra variation which cannotbe handled by the multinomial distribution.
* S-plus/R codes are featured along with practical examplesillustrating the methods.
Practitioners and researchers working in areas such as medicalscience, biological science and social science will benefit fromthis book.

Autoren/Hrsg.

Ng, Kai Wang

Tian, Guo-Liang

Tang, Man-Lai

Inhaltsverzeichnis

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 RandomVector.
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 theDirichlet-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.