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Dirichlet and Related Distributions Theory, Methods and Applications

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ISBN-10: 047068819X

ISBN-13: 9780470688199

Edition: 2011

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

List price: $75.95
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Description:

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 of Dirichlet 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.
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Book details

List price: $75.95
Copyright year: 2011
Publisher: John Wiley & Sons, Limited
Publication date: 4/13/2011
Binding: Hardcover
Pages: 336
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.342
Language: English

Preface
Acknowledgments
List of abbreviations
List of symbols
List of figures
List of tables
Introduction
Motivating examples
Stochastic representation and the d operator
Definition of stochastic representation
More properties on the d operator
Beta and inverted beta distributions
Some useful identities and integral formulae
Partial-fraction expansion
Cambanis-Keener-Simons integral formulae
Hermite-Genocchi integral formula
The Newton-Raphson algorithm
Likelihood in missing-data problems
Missing-data mechanism
The expectation-maximization (EM) algorithm
The expectation/conditional maximization (ECM) algorithm
The EM gradient algorithm
Bayesian MDPs and inversion of Bayes' formula
The data augmentation (DA) algorithm
True nature of Bayesian MDP: inversion of Bayes' formula
Explicit solution to the DA integral equation
Sampling issues in Bayesian MDPs
Basic statistical distributions
Discrete distributions
Continuous distributions
Dirichlet distribution
Definition and basic properties
Density function and moments
Stochastic representations and mode
Marginal and conditional distributions
Survival function and cumulative distribution function
Survival function
Cumulative distribution function
Characteristic functions
The characteristic function of u &#8764; U(T<sub>n</sub>)
The characteristic function of v &#8764; U(V<sub>n</sub>)
The characteristic function of a Dirichlet random vector
Distribution for linear function of a Dirichlet random vector
Density for linear function of v &#8764; U(V<sub>n</sub>)
Density for linear function of u &#8764; U(T<sub>n</sub>)
A unified approach to linear functions of variables and order statistics
Cumulative distribution function for linear function of a Dirichlet random vector
Characterizations
Mosimann's characterization
Darroch and Ratcliff's characterization
Characterization through neutrality
Characterization through complete neutrality
Characterization through global and local parameter independence
MLEs of the Dirichlet parameters
MLE via the Newton-Raphson algorithm
MLE via the EM gradient algorithm
Analyzing serum-protein data of Pekin ducklings
Generalized method of moments estimation
Method of moments estimation
Generalized method of moments estimation
Estimation based on linear models
Preliminaries
Estimation based on individual linear models
Estimation based on the overall linear model
Application in estimating ROC area
The ROC curve
The ROC area
Computing the posterior density of the ROC area
Analyzing the mammogram data of breast cancer
Grouped Dirichlet distribution
Three motivating examples
Density function
Basic properties
Marginal distributions
Conditional distributions
Extension to multiple partitions
Density function
Some properties
Marginal distributions
Conditional distributions
Statistical inferences: likelihood function with GDD form
Large-sample likelihood inference
Small-sample Bayesian inference
Analyzing the cervical cancer data
Analyzing the leprosy survey data
Statistical inferences: likelihood function beyond GDD form
Incomplete 2�2 contingency tables: the neurological complication data
Incomplete r � c contingency tables
Wheeze study in six cities
Discussion
Applications under nonignorable missing data mechanism
Incomplete r � c tables: nonignorable missing mechanism
Analyzing the crime survey data
Nested Dirichlet distribution
Density function
Two motivating examples
Stochastic representation, mixed moments, and mode
Marginal distributions
Conditional distributions
Connection with exact null distribution for sphericity test
Large-sample likelihood inference
Likelihood with NDD form
Likelihood beyond NDD form
Comparison with existing likelihood strategies
Small-sample Bayesian inference
Likelihood with NDD form
Likelihood beyond NDD form
Comparison with the existing Bayesian strategy
Applications
Sample surveys with nonresponse: simulated data
Dental caries data
Competing-risks model: failure data for radio transmitter receivers
Sample surveys: two data sets for death penalty attitude
Bayesian analysis of the ultrasound rating data
A brief historical review
The neutrality principle
The short memory property
Inverted Dirichlet distribution
Definition through the density function
Density function
Several useful integral formulae
The mixed moment and the mode
Definition through stochastic representation
Marginal and conditional distributions
Cumulative distribution function and survival function
Cumulative distribution function
Survival function
Characteristic function
Univariate case
The confluent hypergeometric function of the second kind
General case
Distribution for linear function of inverted Dirichlet vector
Introduction
The distribution of the sum of independent gamma variates
The case of two dimensions
Connection with other multivariate distributions
Connection with the multivariate t distribution
Connection with the multivariate logistic distribution
Connection with the multivariate Pareto distribution
Connection with the multivariate Cook-Johnson distribution
Applications
Bayesian analysis of variance in a linear model
Confidence regions for variance ratios in a linear model with random effects
Dirichlet-multinomial distribution
Probability mass function
Motivation
Definition via a mixture representation
Beta-binomial distribution
Moments of the distribution
Marginal and conditional distributions
Marginal distributions
Conditional distributions
Multiple regression
Conditional sampling method
The method of moments estimation
Observations and notations
The traditional moments method
Mosimann's moments method
The method of maximum likelihood estimation
The Newton-Raphson algorithm
The Fisher scoring algorithm
The EM gradient algorithm
Applications
The forest pollen data
The teratogenesis data
Testing the multinomial assumption against the Dirichlet-multinomial alternative
The likelihood ratio statistic and its null distribution
The C(�) test
Two illustrative examples
Truncated Dirichlet distribution
Density function
Definition
Truncated beta distribution
Motivating examples
Case A: matrix � is known
Case B: matrix � is unknown
Case C: matrix � is partially known
Conditional sampling method
Consistent convex polyhedra
Marginal distributions
Conditional distributions
Generation of random vector from a truncated Dirichlet distribution
Gibbs sampling method
The constrained maximum likelihood estimates
Application to misclassification
Screening test with binary misclassifications
Case-control matched-pair data with polytomous misclassifications
Application to uniform design of experiment with mixtures
Other related distributions
The generalized Dirichlet distribution
Density function
Statistical inferences
Analyzing the crime survey data
Choice of an effective importance density
The hyper-Dirichlet distribution
Motivating examples
Density function
The scaled Dirichlet distribution
Two motivations
Stochastic representation and density function
Some properties
The mixed Dirichlet distribution
Density function
Stochastic representation
The moments
Marginal distributions
Conditional distributions
The Liouville distribution
The generalized Liouville distribution
Some useful S-plus Codes
References
Author index
Subject index