Ridge regression is a well-known technique to estimate the coefficients of a linear model. The method of regularization is a similar approach commonly used to solve underdetermined linear equations with discrete noisy data. When applying such a technique, the choice of the smoothing (or regularization) parameter $h$ is crucial. Generalized cross-validation (GCV) and Mallows' $C_L$ are two popular methods for estimating a good value for $h,$ from the data. Their asymptotic properties, such as consistency and asymptotic optimality, have been largely studied [Craven and Wahba (1979); Golub, Heath and Wahba (1979); Speckman (1985)]. Very interesting convergence results for the actual (random) parameter given by GCV and $C_L$ have been shown by Li (1985, 1986). Recently, Girard (1987, 1989) has proposed fast randomized versions of GCV and $C_L.$ The purpose of this paper is to show that the above convergence results also hold for these new methods.
Publié le : 1991-12-14
Classification:
GCV,
$C_L$,
ridge regression,
regularization,
smoothing splines,
Monte Carlo techniques,
randomized versions,
asymptotic optimality,
62G05,
65U05,
65D10,
65R20,
92A07
@article{1176348380,
author = {Girard, Didier A.},
title = {Asymptotic Optimality of the Fast Randomized Versions of GCV and $C\_L$ in Ridge Regression and Regularization},
journal = {Ann. Statist.},
volume = {19},
number = {1},
year = {1991},
pages = { 1950-1963},
language = {en},
url = {http://dml.mathdoc.fr/item/1176348380}
}
Girard, Didier A. Asymptotic Optimality of the Fast Randomized Versions of GCV and $C_L$ in Ridge Regression and Regularization. Ann. Statist., Tome 19 (1991) no. 1, pp. 1950-1963. http://gdmltest.u-ga.fr/item/1176348380/