On a topic of generalized linear mixed models and stochastic volatility model

by Yam, Ho-kwan

(Uncorrected OCR) Abstract of the thesis entitled On a Topic of Generalized Linear Mixed Models and Stochastic Volatility Model submitted by Yam, Ho Kwan for the degree of Master of Philosophy at The University of Hong Kong in October 2002 Generalized Linear Mixed Models (GLMMs) are extensions to the Generalized Linear Models (GLMs). The inclusion of random effects into the models widens the scope of applicability of GLMMs considerably. However, it also increases the computational effort in estimation. There are various methodologies in making inference on the GLMMs nowadays. We attempt to investigate the use of Gibbs output within the Bayesian framework to carry out the Monte Carlo Approximation of the complicated likelihood function involving random effects by a classical likelihood approach. This methodology is a combination of classical likelihood and Bayesian approaches and it serves as a bridge between them. We will demonstrate this methodology using a famous Salamander Mating data reported by McCullagh and Nelder (1989). Moreover, although the normal distribution plays an important role in statistics, it is not suitable for modeling GLMMs with outlying random effects. The use of a general class of random effects models, such as heavy-tailed distributions should be considered. We will further investi- i gate the use of the Student-i distribution on the random effects in the Salamander Mating data. Furthermore, we suggest to adopt a scale mixture of normal form on the Student-i distribution for simplification of calculation and at the same time, locating the outliers directly through the mixing parameters. Nevertheless, since most financial and economic data exhibits a thick tail behavior, we further investigate the use of heavy tail distributions instead of normal distribution on these kinds of data. A two stage hierarchical scale mixture form on the Student-^ distribution will be demonstrated on the Stochastic Volatility (SV) model in a Bayesian aspect. it