Minimax estimators of the multinormal mean: Autoregressive priors

Empirical Bayes estimators are given for the mean of a k-dimensional normal distribution, k ≥ 3. We assume that y ～ N_k(θ, V₁), V₁ = diag(v_i), v_i known (i = 1, 2,…, k); also, θ ～ N_k(0, V₂) ? V₂ defined by one or more unknown parameters. Of particular interest is V₂ 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 V₂. 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.