Improved curve estimation with smoothing splines through local cross-validation

摘要

Much attention has been given recently to the nonparametric estimation of regression functions (or curves). Various methods have been shown to yield consistent estimators of regression curves. But in addition to the estimator itself, one would like an idea of its variability in order to construct confidence intervals or hypothesis tests for the regression curve. Using a Bayesian correspondence between stochastic processes and smoothing splines, pointwise confidence intervals for the regression curve have been developed and given a frequency interpretation using globally cross-validated smoothing spline (GCVSS) estimation. These pointwise confidence intervals hold on average but not uniformly for all points of a regression curve, at their desired confidence level 1 $-$ $alpha$. One would like to have pointwise confidence intervals that hold at the desired confidence level uniformly for all points so that inferences are valid, independently of location on the regression curve.;This lack of uniformity is due to the changing curvature of the regression curve. The curvature of a smoothing spline estimator is determined by its smoothing parameter. The globally cross-validated smoothing spline (GCVSS) estimator reproduces the global curvature of the regression curve but is insensitive to local changes in curvature. At those points where large changes in curvature occur, biases in estimation results and cause the confidence levels of the corresponding pointwise confidence intervals to be much lower than desired. To deal with this problem, a new smoothing spline estimator has been developed. This new estimator uses a local cross-validation criterion to determine a smoothing parameter for each point. The smoothing parameters are then used to determine the point estimators of the regression curve and the corresponding pointwise confidence intervals. To determine the local cross-validation criterion, a local weighting scheme is used around the point to be estimated. Using a heuristic argument, an empirical choice for the weighting scheme is suggested to be of the same order of n as the asymptotically optimal choice for the weighting scheme. Incorporation of local information through this local cross-validation aspect will be shown to yield uniformly valid pointwise confidence intervals for regression curves.