Rates in the empirical bayes estimation problem with non-identical components

Abstract

This paper considers empirical Bayes estimation of the mean θ of the univariate normal densityf ₀ with known variance where the sample sizesm(n) may vary with the component problems but remain bounded by$\bar m$<∞. Let {(θ _n ,X _n =(X _n,1,...,X _{n, m(n)} ))} be a sequence of independent random vectors where theθ _n are unobservable and iidG and, givenθ _n =θ has densityf ^m(n) _θ . The first part of the paper exhibits estimators for the density of$\sum\limits_{j = 1}^{m(n)} {X_{n,j} }$ and its derivative whose mean-squared errors go to zero with rates$O(n^{ - 1/\bar m} \log n)$ and$O\left( {n^{ - 1/\bar m} (\log n)^2 } \right)$ respectively. LetR ^m(n+1)(G) denote the Bayes risk in the squared-error loss estimation ofθ _n+1 usingX _n+1. For given 0<a<1, we exhibitt _n (X₁,...,X _n ;X _n+1) such that$D(t_n ,G) = E[(t_n - \theta _{n + 1} )^2 ] - R^{m(n + 1)} (G) \leqq c_1 (a,\bar m)(\log n)^2$.$n^{ - a/((2 + a)\bar m)}$ forn>1 under the assumption that the support ofG is in [0, 1]. Under the weaker condition that E[|θ|^2+γ]<∞ for some γ>0, we exhibitt ^* _n (X ₁,...,X _n ;X _n+1) such that$D(t_n^* ,G) \leqq c_2 (\bar m,\gamma )(\log n)^{ - \gamma /(2 + \gamma )}$ forn>1.