OPTIMAL ROBUST BAYES ESTIMATION

A robust viewpoint is to make inferences which are insensitive to the subjective information that can be quantified in terms of a class Gamma of possible prior distributions. The problems of obtaining an optimal robust action, such as an optimal robust credible set or an optimal robust point estimator, are discussed using the Gamma-posterior expected loss as the criterion. The main concern in the paper is the optimal robust estimation of an unknown mean a of a normal distribution with or without the knowledge of the variance sigma(2). It is assumed that the prior distribution of a is in a conjugate prior class. Under the 0-1 loss function, the optimal robust Gamma-posterior credible set is obtained and under the squared error loss function, the optimal robust Gamma-posterior point estimate and the associated confidence interval of theta are obtained. It is shown that the optimal robust Gamma-posterior point estimator of theta has the same asymptotic normal distribution as the MLE for a when the sampling variance approaches zero.