Abstract |
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This paper deals with the following approach for estimating the mean mu of an n-dimensional random vector Y: first, a family S of n x n matrices is specified. Then, an element S is an element of S is selected by Mallows C-L, and mu = 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)parallel to mu - S.Y parallel to(2)(2) - (1/n)parallel to mu - S mu.Y parallel to(2)(2) where S mu denotes the minimizer of (1/n)parallel to mu - S.Y parallel to(2)(2) for S is an element of 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 C-L. 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. |