Adaptive wavelet empirical Bayes estimation of a location or a scale parameter |
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Authors: | Marianna Pensky |
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Affiliation: | Department of Mathematics, University of Central Florida, P.O. Box 161364, Orlando FL 32816, USA |
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Abstract: | Assume that in independent two-dimensional random vectors (X1,θ1),…,(Xn,θn), each θi is distributed according to some unknown prior density function g. Also, given θi=θ, Xi has the conditional density function q(x−θ), x,θ(−∞,∞) (a location parameter case), or θ−1q(x/θ), x,θ(0,∞) (a scale parameter case). In each pair the first component is observable, but the second is not. After the (n+1)th pair (Xn+1,θn+1) is obtained, the objective is to construct an empirical Bayes (EB) estimator of θ. In this paper we derive the EB estimators of θ based on a wavelet approximation with Meyer-type wavelets. We show that these estimators provide adaptation not only in the case when g belongs to the Sobolev space H with an unknown , but also when g is supersmooth. |
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Keywords: | Empirical Bayes estimation Adaptive estimation Meyer-type wavelet Posterior and prior risk |
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