Consistency of Posterior Mixtures in the Gaussian Family on a Hilbert Space and Its Applications

Sumon Majumdar

Journal of Multivariate Analysis, 1995, vol. 55, issue 2, 187-204

Abstract: Majumdar (1994, J. Multivariate Anal.48 87-105) compounds (in the sense of Robbins, 1951, "Proceedings, Second Berkeley Sympos. Math. Statist. Probab.," pp. 131-148, Univ. of California Press, Berkeley) the estimation problem in the mean-parameter family of Gaussian distributions on a real separable infinite dimensional Hilbert space. The question of asymptotic optimality of compound estimators that are Bayes versus a hyperprior mixture of i.i.d. priors on the compound parameter is reduced there, under a compactness restriction on the parameter space, to the question of consistency, in an extended sense, of a certain posterior mixture for the empirical mixture. For mixing hyperpriors with full topological support, that consistency result is obtained in this paper. A corollary of the consistency result is applied to obtain asymptotically optimal decision rules in the empirical Bayes problem involving the mean-parameter Gaussian family and a sufficiently smooth risk function.

Date: 1995
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