Dirichlet and related distributions: theory, methods and applications

This book provides a comprehensive review on the Dirichlet distribution including its basic properties, marginal and conditional distributions, cumulative distribution and survival functions. The authors provide insight into new materials such as survival function, characteristic functions for two uniform distributions over the hyper-plane and simplex distribution for linear function ofDirichlet components estimation via the expectation-maximization gradient algorithm and application. Two new families of distributions (GDD and NDD) are explored, with emphasis on applications in incomplete categorical data and survey data with non-response. Theoretical results on inverted Dirichlet distribution and its applications are featured along with new results that deal with truncated Dirichlet distribution, Dirichlet process and smoothed Dirichlet distribution. The final chapters look at results gathered for Dirichlet-multinomial distribution, Generalized Dirichlet distribution, Liouville distribution, generalized Liouville distribution and matrix-variate Dirichlet distribution. INDICE: Preface. Acknowledgments. List of Figures. List of Tables. List ofAbbreviations. 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 ofMoments Estimation. 2.9 Estimation Based on Linear Models. 2.10 Application in Estimating Receiver Operating Characteristic (ROC) Area. 3 Grouped DirichletDistribution. 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 Functionwith 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. 4.5Conditional 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 Distributio

• ISBN: 978-0-470-68819-9
• Editorial: John Wiley & Sons
• Encuadernacion: Cartoné
• Páginas: 336
• Fecha Publicación: 15/04/2011
• Nº Volúmenes: 1
• Idioma: Inglés