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Journal Article

# Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter

Gene H. Golub, Michael Heath and Grace Wahba
Technometrics
Vol. 21, No. 2 (May, 1979), pp. 215-223
DOI: 10.2307/1268518
Stable URL: http://www.jstor.org/stable/1268518
Page Count: 9

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## Abstract

Consider the ridge estimate β̂(λ) for β in the model y=Xβ +ε,ε ∼ N(0,σ 2I), σ 2 unknown, β̂(λ)=(XTX+nλ I)-1 XTy. We study the method of generalized cross-validation (GCV) for choosing a good value λ̂ for λ, from the data. The estimate λ̂ is the minimizer of V(λ) given by $V(\lambda)=\frac{1}{n}\|(I-A(\lambda))y\|^{2}/\left[\frac{1}{n}\text{Trace}(I-A(\lambda))\right]^{2}$, where A(λ)=X(XTX+nλ I)-1XT. This estimate is a rotation-invariant version of Allen's PRESS, or ordinary cross-validation. This estimate behaves like a risk improvement estimator, but does not require an estimate of σ 2, so can be used when n - p is small, or even if p ≥ n in certain cases. The GCV method can also be used in subset selection and singular value truncation methods for regression, and even to choose from among mixtures of these methods.

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