Bayes compound and empirical Bayes estimation of the mean of a Gaussian distribution on a Hilbert space

The problem of finding admissible and asymptotically optimal (in the sense of Robbins) compound and empirical Bayes rules is investigated, when the component problem is estimation of the mean of a Gaussian distribution (with a known one-to-one covariance C) on a real separable infinite dimensional Hilbert space H under weighted Squared-Error-Loss. The parameter set is restricted to be a compact subset of the Hilbert space isomorphic to H via C1/2. We note that all Bayes compound estimators in our problem are admissible. Our main result is that those Bayes versus a mixture of i.i.d. priors on the compound parameter are a.o. if the mixing hyperprior has full support. The same result holds in the empirical Bayes formulation as well.