Abstract.    Let {(X_i,theta_i)} be a sequence of independent random vectors where X_i, conditional on theta_i, has the probability density of the form f(x | theta_i) = u(x)C(theta_i)exp(-x/theta_i) and the unobservable theta_i are i.i.d. according to an unknown G in some class \cal G of prior distributions on Theta, a subset of {theta > 0 | C(theta) = \big( \int u(x)exp(-x/theta)dx\big)^-1 > 0}. For a \cal S(X₁,....,X_n,X_n+1)-measurable function phi_n, let R_n = E(phi_n-theta_n+1)² denote the Bayes risk of phi_n and let R(G) denote the infimum Bayes risk with respect to G. For each integer s > 1 we exhibit a class of \cal S(X₁,....,X_n,X_n+1)-measurable functions phi_n such that for delta in [s^-1,1], c₀n^-2s/(1+2s) < R_n(phi_n,G)-R(G)< c₁n^{-2(sdelta-1)/(1+2s)} under certain conditions on u and G. No assumptions on the form or smoothness of u is made, however. Examples of functions u, including one with infinitely many discontinuities, are given for which our conditions reduce to some moment conditions on G. When Theta is bounded, for each integer s > 1 \cal S(X₁,....,X_n,X_n+1)-measurable functions phi_n are exhibited such that for delta in [2/s,1], c'₀n^-2s/(1+2s) < R(phi_n,G)-R(G) < c'₁n^{-2sdelta/(1+2s)}. Examples of functions u and class \cal G are given where the above lower and upper bounds are achieved.

Key words and phrases:    Exponential family, empirical Bayes, estimation, asymptotic optimality, rates and best rates.

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