We consider estimating the infinite dimensional normal means from the prior Bayes standpoint and study the robustness properties of the prior Bayes procedure. Assumptions are made that the prior variances of the parameter vector are decreasing, with some extension. Under squared error loss, the empirical Bayes type estimation of the parameter vector is derived without calculating the posterior. The robustness of our procedure is justified by estimating the increase of prior risk over the optimal procedure; simulations indicate that this is valid for reasonable sample sizes. Small relaxations in the prior assumption still preserve the asymptotic property. Similar approach is then applied to estimate the spectral densities of stationary processes with the corresponding prior assumptions made on the autocovariance function. We show by simulation that our procedure, which is barely even somewhat unreasonable prior Bayes, compares favorably with many procedures now in wide use. Furthermore, our method for the linear ordering of parameters can be generalized to other partial orderings, such as obtaining estimates of spectral densities for stationary spatial or spatio-temporal processes.
