The Annals of Statistics

From Stein's Unbiased Risk Estimates to the Method of Generalized Cross Validation

Ker-Chau Li

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Abstract

This paper concerns the method of generalized cross validation (GCV), a promising way of choosing between linear estimates. Based on Stein estimates and the associated unbiased risk estimates (Stein, 1981), a new approach to GCV is developed. Many consistency results are obtained for the cross-validated (Steinized) estimates in the contexts of nearest-neighbor nonparametric regression, model selection, ridge regression, and smoothing splines. Moreover, the associated Stein's unbiased risk estimate is shown to be uniformly consistent in assessing the true loss (not the risk). Consistency properties are examined as well when the sampling error is unknown. Finally, we propose a variant of GCV to handle the case that the dimension of the raw data is known to be greater than that of their expected values.

Article information

Source
Ann. Statist. Volume 13, Number 4 (1985), 1352-1377.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176349742

Digital Object Identifier
doi:10.1214/aos/1176349742

Mathematical Reviews number (MathSciNet)
MR811497

Zentralblatt MATH identifier
0605.62047

JSTOR
links.jstor.org

Subjects
Primary: 62G99: None of the above, but in this section
Secondary: 62J99: None of the above, but in this section 62J05: Linear regression 62J07: Ridge regression; shrinkage estimators

Keywords
Cross validation generalized cross validation model selection nearest-neighbor estimates ridge regression smoothing spline Stein estimates Stein's unbiased risk estimates

Citation

Li, Ker-Chau. From Stein's Unbiased Risk Estimates to the Method of Generalized Cross Validation. Ann. Statist. 13 (1985), no. 4, 1352--1377. doi:10.1214/aos/1176349742. http://projecteuclid.org/euclid.aos/1176349742.


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