Summary: This paper deals with the following approach for estimating the mean of an -dimensional random vector : first, a family of matrices is specified. Then, an element is selected by Mallows , and . The case is considered that 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
where denotes the minimizer of for . These probability bounds are generalized to the situation that is the union of a moderate number of ordered linear smoothers.
These results complement work by K. C. Li [ibid. 14, 1101-1112 (1986; Zbl 0629.62043); ibid. 15, 958-975 (1987; Zbl 0653.62037)] on the asymptotic optimality of . 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.