ADJUSTMENT FOR DENSITY METHOD TO ESTIMATE RANDOM EFFECTS IN HIERARCHICAL BAYES MODELS

Abstract

The Adjustment for Density Method (ADM) has received considerable attention in recent years. The method was proposed about thirty years back in approximating a complex univariate density by a density from the Pearson family of distributions. The ADM has been developed to approximate posterior distributions of hyper-parameters, shrinkage parameters and random effects of a few well-known univariate hierarchical Bayesian models. This dissertation advances the ADM to approximate posterior distributions of hyper-parameters, shrinkage parameters, synthetic probabilities and multinomial probabilities associated with a multinomial-Dirichlet-logit Bayesian hierarchical model. The method is adapted so it can be applied to weighted counts. We carefully propose prior for the hyper-parameters of the multinomial-Dirichlet-logit model so as to ensure propriety of posterior of relevant parameters of the model and to achieve good small sample properties. Following general guidelines of the ADM for univariate distributions, we devise suitable adjustments to the posterior density of the hyper-parameters so that adjusted posterior modes lie in the interior of the parameter space and to reduce the bias in the point estimates. Beta distribution approximations are employed when approximating the posterior distributions of the individual shrinkage factors and Dirichlet distribution approximations are used when approximating the posterior distributions of the synthetic probabilities. The parameters of the beta or the Dirichlet posterior density are approximated carefully so the method approximates the exact posterior densities accurately. Simulation studies demonstrate that our proposed approach in estimating the multinomial probabilities in the multinomial-Dirichlet-logit model is accurate in estimation, fast in speed and has better operating characteristics compared to other existing procedures. We consider two applications of our proposed hierarchical Bayes model using complex survey and Big Data. In the first example, we consider small area gender proportions using a binomial-beta-logit model. The proposed method improves on a rival method in terms of smaller margins of error. In the second application, we demonstrate how small area multi-category race proportions estimates, obtained by direct method applied on Twitter data, can be improved by the proposed method. This dissertation ends with a discussion on future research in the area of ADM.