CHAPTER 8 - Decision Theoretic and Bayesian Approach in Testing and Estimation

Abstract

The chapter discusses decision theoretic and Bayesian approach in testing and estimation. The decision theoretic approach provides a general framework for both estimation of parameters and testing hypotheses. The risk functions depend on the parameters θ of the parent distribution. The average risk can be defined as an expected risk according to some probability distribution on the parameter space. This expected risk is called in Bayesian theory the prior risk and the probability measure on the parameter space is called a prior distribution. The estimators or test functions that minimize the prior risk, with respect to some prior distribution, are called Bayes procedures for the specified prior distribution. Bayes procedures have certain desirable properties. The chapter is describes the structure of optimal decision rules in the framework of Bayesian theory. In the Bayesian approach, the unknown parameters are considered as values determined at random according to some specified distribution, called the prior distribution. This prior distribution can be conceived as a normalized nonnegative weight function that the statistician assigns to the various possible parameter values.