Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

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
  • Download ($14.00)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter
Preview not available

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.

Page Thumbnails

  • Thumbnail: Page 
215
    215
  • Thumbnail: Page 
216
    216
  • Thumbnail: Page 
217
    217
  • Thumbnail: Page 
218
    218
  • Thumbnail: Page 
219
    219
  • Thumbnail: Page 
220
    220
  • Thumbnail: Page 
221
    221
  • Thumbnail: Page 
222
    222
  • Thumbnail: Page 
223
    223