Uniform scale mixture models with applications to Bayesian inference.

Abstract

This dissertation is on scale mixture models and their applications to Bayesian inference. It focuses on two main themes: (1) <italic>Modeling </italic>: Develop a general family of statistical models using scale mixtures of uniform distributions. The main attractive feature of such a family is that it enables the incorporation of difficult, but realistic, assumptions in a broad range of applications. (2) <italic>Computation</italic>: The development of uniform scale mixture models has a secondary merit; it makes the analyses of real data straightforward by an appropriate use of auxiliary (or latent) variables in the resulting computational form of the model. Using the theoretical developments outlined above, the thesis then focuses on a variety of applications, particularly from a Bayesian perspective. As illustrations, consider the following. (A) <italic>Robust modeling</italic>: Most of the currently used parametric models assume error terms to have normal distributions, which in many situations may not be realistic. If deviations from normality are entertained the problem of parameter estimation, Bayesian or otherwise, tends to get complicated. The thesis demonstrates that using the scale mixture characterization, difficulties in estimation are straightforwardly obviated even if one deviates from normal models. From a practitioner's perspective, this leads to a robust analysis of the data. (B) <italic>Variance regression </italic>: By using scale mixture of uniform methodology, one is able to estimate both the mean and variance parameters simultaneously. Traditionally, the estimation is carried out separately. (C) <italic>Other models</italic>: The flexibility of scale mixture of uniforms is also demonstrated in estimating parameters in simultaneous equation models. The thesis undertakes a detailed study of such models. Also, the class of Box-Cox models and models involving stable laws are tackled using the theoretical methods developed in this thesis.