LOCALLY ADAPTIVE WAVELET EMPIRICAL BAYES ESTIMATION OF A LOCATION PARAMETER

MARIANNA PENSKY

Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, U.S.A.

(Received December 10, 1999; revised October 18, 2000)

Abstract. The traditional empirical Bayes (EB) model is considered with the parameter being a location parameter, in the situation when the Bayes estimator has a finite degree of smoothness and, possibly, jump discontinuities at several points. A nonlinear wavelet EB estimator based on wavelets with bounded supports is constructed, and it is shown that a finite number of jump discontinuities in the Bayes estimator do not affect the rate of convergence of the prior risk of the EB estimator to zero. It is also demonstrated that the estimator adjusts to the degree of smoothness of the Bayes estimator, locally, so that outside the neighborhoods of the points of discontinuities, the posterior risk has a high rate of convergence to zero. Hence, the technique suggested in the paper provides estimators which are significantly superior in several respects to those constructed earlier.

Key words and phrases: Empirical Bayes estimation, adaptive estimation, wavelet, posterior and prior risks.