Estimation of Discrete Distributions with A Class of Simplex Constraints

(PREVIOUS TITLE: ESTIMATION OF DISCRETE DISTRIBUTIONS WITH SIMPLEX CONSTRAINTS AND ITS APPLICATIONS) Simplex constraints, such as monotonicity and convex constraints, on the probabilities of a set of discrete distributions are useful for modeling and analyzing discrete data. For example, they provide an excellent class of alternatives to saturated models for exploratory data analysis in general and for multiple imputation in particular. This paper gives a formal definition of simplex constraints and focuses on but is not limited to monotonicity and convex/concave constraints on binomial, Poisson, and multinomial distributions. We present a general framework for both maximum likelihood estimation and Bayesian estimation of discrete distribution with simplex constraints using the EM algorithm and the DA algorithm and most importantly, provide detailed formulation and implementation of EM and DA for binomial, Poisson, Poisson-binomial, multinomial and hierarchical multinomial distributions. Applications of these models are illustrated with numerical examples. Other potential applications of these models are also addressed along with a brief discussion on ways of constructing simplex constraints via model expansion, including both data augmentation and parameter expansion.