Asymptotic Solutions to the Two State Component Compound Decision Problem, Bayes Versus Diffuse Priors on Proportions

Abstract

Gilliland and Hannan (1974, Section 3) consider a general finite state compact risk component and reduce the problem of treating the asymptotic excess compound risk of Bayes compound rules to the question of $L_1$ consistency of certain induced estimators. This present paper considers the two state case and for several classes of diffuse symmetric priors on proportions establishes the $L_1$ consistency with rate. The rate $O(n^{-\frac{1}{2}})$ uniform in state sequences is shown for the uniform prior giving strong affirmation to the asymptotic form of a conjecture by Robbins (1951). The same or logarithmically weakened rate is shown for symmetric priors which are $\Lambda$-mixtures for several classes of $\Lambda$. A corollary shows a nonnull consistency, without regularity conditions, of a maximum likelihood estimator.