The nested dirichlet distribution and incomplete categorical data analysis

Abstract

The nested Dirichlet distribution (NDD) is an important distribution defined on the closed n-dimensional simplex. It includes the classical Dirichlet distribution and is useful in incomplete categorical data (ICD) analysis. In this article, we develop the distributional properties of NDD. New large-sample likelihood and small-sample Bayesian approaches for analyzing ICD are proposed and compared with existing likelihood/Bayesian strategies. We show that the new approaches have at least three advantages over existing approaches based on the traditional Dirichlet distribution in both frequentist and conjugate Bayesian inference for ICD. The new methods possess closed-form expressions for both the maximum likelihood and Bayes estimates when the likelihood function is in NDD form; produce computationally efficient EM and data augmentation algorithms when the likelihood is not in NDD form; and provide exact sampling procedures for some special cases. The methodologies are illustrated with simulated and real data.