Bayesian Estimation of Multivariate Location Parameters

This paper presents an expository development of Bayesian estimation with substantial emphasis on exact results for the multivariate normal location models with respect to squared error loss. From the time Stein, in 1956, showed the inadmissibility of the best invariant estimator when sampling from a multivariate normal distribution in 3 or more dimensions, there has been an outpouring of improved estimators with a Bayesian flavor, encouraged largely by the connections between Bayes estimation, admissibility and minimaxity. In this chapter, we attempt to give a coherent presentation of numerous Bayesian results (proper, generalized, empirical) for this case. Generalizations for the location parameter of multivariate normal distributions with unknown covariance matrices and general quadratic loss are also presented