| |

| |

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 ∼ U(T<sub>n</sub>) | |

| |

| |

| |

The characteristic function of v ∼ 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 ∼ U(V<sub>n</sub>) | |

| |

| |

| |

Density for linear function of u ∼ 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 2ï¿½2 contingency tables: the neurological complication data | |

| |

| |

| |

Incomplete r ï¿½ c contingency tables | |

| |

| |

| |

Wheeze study in six cities | |

| |

| |

| |

Discussion | |

| |

| |

| |

Applications under nonignorable missing data mechanism | |

| |

| |

| |

Incomplete r ï¿½ 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(ï¿½) test | |

| |

| |

| |

Two illustrative examples | |

| |

| |

| |

Truncated Dirichlet distribution | |

| |

| |

| |

Density function | |

| |

| |

| |

Definition | |

| |

| |

| |

Truncated beta distribution | |

| |

| |

| |

Motivating examples | |

| |

| |

| |

Case A: matrix ï¿½ is known | |

| |

| |

| |

Case B: matrix ï¿½ is unknown | |

| |

| |

| |

Case C: matrix ï¿½ 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 | |