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June, 1994 Ordered Linear Smoothers
Alois Kneip
Ann. Statist. 22(2): 835-866 (June, 1994). DOI: 10.1214/aos/1176325498


This paper deals with the following approach for estimating the mean $\mu$ of an $n$-dimensional random vector $Y$: first, a family $\mathbf{S}$ of $n \times n$ matrices is specified. Then, an element $\widehat{S} \in \mathbf{S}$ is selected by Mallows $C_L$, and $\widehat{\mu} = \widehat{S}\cdot Y$. The case is considered that $\mathbf{S}$ 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 $(1/n)\|\mu - \widehat{S}\cdot Y\|^2_2 - (1/n)\|\mu - \widehat{S}_\mu\cdot Y\|^2_2$, where $\widehat{S}_\mu$ denotes the minimizer of $(1/n)\|\mu - S\cdot Y\|^2_2$ for $S \in \mathbf{S}$. These probability bounds are generalized to the situation that $\mathbf{S}$ 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.


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Alois Kneip. "Ordered Linear Smoothers." Ann. Statist. 22 (2) 835 - 866, June, 1994.


Published: June, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0815.62022
MathSciNet: MR1292543
Digital Object Identifier: 10.1214/aos/1176325498

Primary: 62G07
Secondary: 62J07

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 2 • June, 1994
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