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.
Publié le : 1975-01-14
Classification:
Squared error,
empirical Bayes estimation,
kernel function,
rate of convergence,
asymptotically optimal,
62F15,
62F10
@article{1176343005,
author = {Lin, Pi-Erh},
title = {Rates of Convergence in Empirical Bayes Estimation Problems: Continuous Case},
journal = {Ann. Statist.},
volume = {3},
number = {1},
year = {1975},
pages = { 155-164},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343005}
}
Lin, Pi-Erh. Rates of Convergence in Empirical Bayes Estimation Problems: Continuous Case. Ann. Statist., Tome 3 (1975) no. 1, pp. 155-164. http://gdmltest.u-ga.fr/item/1176343005/