IMPROVING PENALIZED LEAST SQUARES THROUGH ADAPTIVE SELECTION OF PENALTY AND SHRINKAGE

RUDOLF BERAN

Department of Statistics, University of California, Davis, Davis, CA 95616, U.S.A.

(Received February 7, 2001; revised July 2, 2001)

Abstract. Estimation of the mean function in nonparametric regression is usefully separated into estimating the means at the observed factor levels—a one-way layout problem—and interpolation between the estimated means at adjacent factor levels. Candidate penalized least squares (PLS) estimators for the mean vector of a one-way layout are expressed as shrinkage estimators relative to an orthogonal regression basis determined by the penalty matrix. The shrinkage representation of PLS suggests a larger class of candidate monotone shrinkage (MS) estimators. Adaptive PLS and MS estimators choose the shrinkage vector and penalty matrix to minimize estimated risk. The actual risks of shrinkage-adaptive estimators depend strongly upon the economy of the penalty basis in representing the unknown mean vector. Local annihilators of polynomials, among them difference operators, generate penalty bases that are economical in a range of examples. Diagnostic techniques for adaptive PLS or MS estimators include basis-economy plots and estimates of loss or risk.

Key words and phrases: Nonparametric regression, one-way layout, adaptation, loss estimator, risk estimator, economical basis, orthogonal polynomial, local annihilator.