GENERALIZED CROSS-VALIDATION AS A METHOD FOR CHOOSING A GOOD RIDGE PARAMETER

Consider the ridge estimate β(λ) for β in the model unknown, (λ) =(XTX + nλI)−1XTy. We study the method of generalized cross-validation(GCV)for choosing a good value for λ from the data. The estimate is the minimizer of V(λ) given by where A(λ)= X(XTX + nλI)−1XTThis 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 σ2so can be used when n − p is small, or even if p ≥ 2 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. © 1979 Taylor & Francis Group, LLC.