Consistency of posterior mixtures in the Gaussian family on a Hilbert space and its applications

JOURNAL OF MULTIVARIATE ANALYSIS

卷:55

Article

关键词: consistency;    posterior;    compound;    empirical Bayes;    asymptotic optimality;    hyperprior;    isonormal map;   

DOI  :  10.1006/jmva.1995.1074

来源: Elsevier

【 摘 要 】

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. (C) 1995 Academic Press, Inc.

【 授权许可】

Free