Loss functions, utility functions and Bayesian sample size determination.

Abstract

This thesis consists of two parts. The purpose of the first part of the research is to obtain Bayesian sample size determination (SSD) using loss or utility function with a linear cost function. A number of researchers have studied the Bayesian SSD problem. One group has considered utility (loss) functions and cost functions in the SSD problem and others not. Among the former most of the SSD problems are based on a symmetrical squared error (SE) loss function. On the other hand, in a situation when underestimation is more serious than overestimation or vice-versa, then an asymmetric loss function should be used. For such a loss function how many observations do we need to take to estimate the parameter under study? We consider different types of asymmetric loss functions and a linear cost function for sample size determination. For the purposes of comparison, firstly we discuss the SSD for a symmetric squared error loss function. Then we consider the SSD under different types of asymmetric loss functions found in the literature. We also introduce a new bounded asymmetric loss function and obtain SSD under this loss function. In addition, to estimate a parameter following a particular model, we present some theoretical results for the optimum SSD problem under a particular choice of loss function. We also develop computer programs to obtain the optimum SSD where the analytic results are not possible. In the two parameter exponential family it is difficult to estimate the parameters when both are unknown. The aim of the second part is to obtain an optimum decision for the two parameter exponential family under the two parameter conjugate utility function. In this case we discuss Lindley’s (1976) optimum decision for one 6 parameter exponential family under the conjugate utility function for the one parameter exponential family and then extend the results to the two parameter exponential family. We propose a two parameter conjugate utility function and then lay out the approximation procedure to make decisions on the two parameters. We also offer a few examples, normal distribution, trinomial distribution and inverse Gaussian distribution and provide the optimum decisions on both parameters of these distributions under the two parameter conjugate utility function.