Rates of Convergence in Empirical Bayes Estimation Problems: Continuous Case

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 asympto...

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Bibliographic Details
Published in:The Annals of statistics 1975-01, Vol.3 (1), p.155-164
Main Author: Lin, Pi-Erh
Format: Article
Language:English
Subjects:
62F10
62F15
asymptotically optimal
Bayes estimators
Constructive empiricism
Density estimation
empirical Bayes estimation
Empiricism
Estimators
Grants
kernel function
Kernel functions
Mathematical sequences
Mathematics
Perceptron convergence procedure
rate of convergence
Squared error
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Summary: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-qwith q ≤ 1/3. An example is given and an explicit kernel function is indicated.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1176343005