This paper deals with the following approach for estimating the mean of an n-dimensional random vector Y: first, a family S of n n matrices is specified. Then, an element S S is selected by Mallows CL, and = S Y. The case is considered that S is an "ordered linear smoother" according to some easily interpretable, qualitative conditions. Examples include linear smoothing procedures in nonparametric regression (as, e. g. , smoothing splines, minimax spline smoothers and kernel estimators). Stochastic probability bounds are given for the difference (1/n) \| - S Y\|²₂ - (1/n) \| - S_ Y\|²₂, where S_ denotes the minimizer of (1/n) \| - S Y\|²₂ for S S. These probability bounds are generalized to the situation that S is the union of a moderate number of ordered linear smoothers. The results complement work by Li on the asymptotic optimality of CL. Implications for nonparametric regression are studied in detail. It is shown that there exists a direct connection between James-Stein estimation and the use of smoothing procedures, leading to a decision-theoretic justification of the latter. Further conclusions concern the choice of the order of a smoothing spline or a minimax spline smoother and the rates of convergence of smoothing parameters.
Aloïs Kneip (Wed,) studied this question.