Adaptive wavelet empirical Bayes estimation of a location or a scale parameter
empirical Bayes estimation; adaptive estimation; Meyer-type wavelet; posterior and prior risk; DENSITY-ESTIMATION; CURVE ESTIMATION; RATES; CONVERGENCE; ERROR; Statistics & Probability
Assume that in independent two-dimensional random vectors (X-1, theta(1)),..., (X-n, theta(n)), each theta(i) is distributed according to some unknown prior density function g. Also, given theta(i) = theta, X-i has the conditional density function q(x - theta), x, theta is an element of (-infinity, infinity) (a location parameter case), or 0(-1) q(x/0), x, 0 is an element of (0, infinity) (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, theta(n+1)) is obtained, the objective is to construct an empirical Bayes (EB) estimator of 0. In this paper we derive the EB estimators of theta 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-alpha with an unknown alpha, but also when g is supersmooth. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 62C12; 62G20.
Journal of Statistical Planning and Inference
Pensky, M. (2000). Adaptive wavelet empirical Bayes estimation of a location or a scale parameter. Journal of Statistical Planning and Inference, 90(2). doi: 10.1016/s0378-3758(00)00123-3