Minimax estimators of the multinormal mean: Autoregressive priors

Abstract

Empirical Bayes estimators are given for the mean of a k-dimensional normal distribution, k ≥ 3. We assume that y ∼ N k ( θ, V 1), V 1 = diag( v i ), v i known ( i = 1, 2,…, k); also, θ ∼ N k (0, V 2) − V 2 defined by one or more unknown parameters. Of particular interest is V 2 generated by an autoregressive process. A recent result of Efron and Morris is used to obtain necessary and sufficient conditions for the minimaxity of our estimators. Practical sufficient conditions (for minimaxity) are obtained by exploiting the structure of V 2. Another result shows that our estimators have good Bayesian properties. Estimates of the exact size of Pearson's chi-square test are given in an example; the autoregressive prior is very natural in this situation.