GACV for quantile smoothing splines

Quantile smoothing splines provide nonparametric estimation of conditional quantile functions. Like other nonparametric smoothing techniques, the choice of smoothing parameters considerably affects the performance of quantile smoothing splines. The robust cross-validation (RCV) has been commonly used as a tuning criterion in practice. To explain its success, Oh et al. (J. Roy. Statist. Soc. Ser. A, in press) argued that the RCV curve, as a function of smoothing parameters in quantile smoothing splines, differs from the mean squared error (MSE) curve only by a constant. In this article, we consider an alternative loss function, the generalized comparative Kullback-Leibler distance (GCKL) for the quantile smoothing spline. We argue that RCV is an estimator of GCKL. A disadvantage of RCV is its computational intensity. To reduce the associated computational cost, Nychka et al. (J. Amer. Statist. Assoc. 90 (432) (1995) 117 1) has previously proposed an approximation to RCV, namely ACV. However, we find in our simulations that the ACV-based tuning method will often fail in practice. We first reexainine the theoretical basis for ACV. This exercise enables us to explain the failure of ACV. Then we continue to propose a remedy, the generalized approximate cross-validation (GACV) as a computable proxy for the GCKL. Some preliminary simulations suggest that the GACV score is a good estimate of the GCKL score and that the GACV-based tuning technique compares favorably with both ACV and another commonly used criteria, Schwartz information criterion. A real dataset is examined to illustrate the empirical performance of the proposed method. (c) 2004 Published by Elsevier B.V.