Minimax regret analysis of orthogonal series regression estimation: selection versus shrinkage

Abstract

We consider the problem of orthogonal series regression estimation when no prior information is available about the form of the regression curve. In a decision-theoretic framework we study the finite sample properties of two classes of nonlinear estimators, based either on a selection or on a shrinkage approach. The methods used correspond to the hard and soft thresholding techniques in the context of wavelet estimation. The whole analysis is carried out under the simplifying assumption of normally distributed observations with a common known variance. Use of the minimax regret principle shows the superiority of the optimal data-dependent shrunk estimator over its selection-type analogue. The latter behaves similarly to the minimiser of Mallows' C p -criterion, and requires much larger thresholds than the former. Moreover, the minimax regret thresholds differ substantially from those usually recommended for wavelets. In a simulated data example we illustrate the behaviour of the proposed methods.