Asymptotic optimality of \(C_ L\) and generalized cross-validation in ridge regression with application to spline smoothing. (English) Zbl 0629.62043

Li, Ker-Chau

Ann. Stat. 14, 1101-1112 (1986).

The asymptotic optimality of Mallows’ \(C_ L\) and generalized cross- validation is demonstrated in the setting of ridge regression. An application is made to spline smoothing in nonparametric regression. A counterexample is given to help understand why sometimes GCV may not be asymptotically optimal. The coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of GCV. The proof is based on the connection between GCV and Stein’s unbiased risk estimate.

MSC:

62G05	Nonparametric estimation
62G99	Nonparametric inference

Keywords:

Mallow’s CL; asymptotic optimality; generalized cross-validation; ridge regression; spline smoothing; nonparametric regression; coefficient of variation; eigenvalues of the information matrix; Stein’s unbiased risk estimate