Improved inference in nonparametric regression using L(k)-smoothing splines

Smoothing splines are one of the most popular approaches to nonparametric regression. Wahba (J. Roy. Statist. Sec. Ser. B 40 (1978) 364-372; 45 (1983) 133-150) showed that smoothing splines are also Bayes estimates and used the corresponding prior model to derive interval estimates for the regression function. Although the interval estimates work well on a global basis, they can have poor local properties. The source of this problem is the use of a global smoothing parameter. We introduce the notion of L(k)- smoothing splines. These splines allow for a variable smoothing parameter and can substantially improve local inference.