EMPIRICAL BAYES WITH RATES AND BEST RATES OF CONVERGENCE IN u(x)C(θ) exp(−x/θ)-FAMILY: ESTIMATION CASE

R. S. SINGH^{1} AND LAISHENG WEI^{2}

1Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 2Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, P. R. China

(Received May 30, 1990; revised June 24, 1991)

Abstract. Let {(X_{i}, θ_{i})} be a sequence of independent random vectors where X_{i}, conditional on θ_{i}, has the probability density of the form f(x | θ_{i}) = u(x)C(θ_{i}) exp(−x/θ_{i}) and the unobservable θ_{i} are i.i.d. according to an unknown G in some class G of prior distributions on Θ, a subset of {θ > 0 | C(θ) =

( ∫ u(x) exp(−x/θ)dx)^{−1} > 0}. For a S(X_{1}, …, X_{n}, X_{n+1})-measurable func-

tion ϕ_{n}, let R_{n} = E(ϕ_{n} − θ_{n+1})^{2} denote the Bayes risk of ϕ_{n} and let R(G) denote the infimum Bayes risk with respect to G. For each integer s > 1 we exhibit a class of S(X_{1}, …, X_{n}, X_{n+1})-measurable functions ϕ_{n} such that for δ in [s^{−1}, 1], c_{0}n^{−2s/(1+2s)} ≤ R_{n}(ϕ_{n}, G) − R(G) ≤ c_{1}n^{−2(sδ−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 Θ is bounded, for each integer s > 1 S(X_{1}, …, X_{n}, X_{n+1})-measurable functions ϕ_{n} are exhibited such that for δ in [2/s, 1], c′_{0}n^{−2s/(1+2s)} ≤ R(ϕ_{n}, G) − R(G) ≤ c′_{1}n^{−2sδ/(1+2s)}. Examples of functions u and class 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.