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

✍ Scribed by S. Majumdar

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 555 KB
• Volume: 55
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1995.1074

✦ Synopsis

Majumdar (1994, J. Multivariate Anal. 48 87-106) 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. (\quad 1995) Academic Press, Inc: