Rates of Convergence in Empirical Bayes Estimation Problems: Continuous Case
The Annals of Statistics · 1975 · Volume 3 · Issue 1
Abstract
In this paper we construct sequences of estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions. Utilizing these estimators, a sequence of empirical Bayes estimators is proposed. It is found that this sequence is asymptotically optimal in the sense of Robbins (Ann. Math. Statist. 35 (1964) 1-20). The rates of convergence of the Bayes risks associated with the proposed empirical Bayes estimators are obtained. It is noted that the exact rate is $n^{-q}$ with $q \leqq \frac{1}{3}$. An example is given and an explicit kernel function is indicated.
Keywords
62F10, 62F15, Asymptotically optimal, empirical Bayes estimation, kernel function, rate of convergence, Squared error
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