Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

    Authors

    R. Benhaddou;M. Pensky

    Comments

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    Abbreviated Journal Title

    J. Stat. Plan. Infer.

    Keywords

    Adaptivity; Empirical Bayes estimation; Wavelets; Convergence rate; EXPONENTIAL-FAMILIES; CONVERGENCE-RATES; PARAMETER; POPULATION; LOCATION; TESTS; Statistics & Probability

    Abstract

    In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples. (C) 2013 Elsevier B.V. All rights reserved.

    Journal Title

    Journal of Statistical Planning and Inference

    Volume

    143

    Issue/Number

    10

    Publication Date

    1-1-2013

    Document Type

    Article

    Language

    English

    First Page

    1672

    Last Page

    1688

    WOS Identifier

    WOS:000323809800007

    ISSN

    0378-3758

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