On spline estimators and prediction intervals in nonparametric regression

摘要

The quantile regression function gives the quantile in the conditional distribution of a response variable given the value of a covariate. It can be used to measure the effect of covariates not only in the center of a population, but also in the upper and lower tails. Moreover, it provides prediction intervals that do not rely on normality or other distributional assumptions. In a nonparametric setting, we explore a class of quantile regression spline estimators of the quantile regression function. We consider an automatic knot selection procedure involving a linear programming method, stepwise knot addition using a modified AIC, and stepwise knot deletion using a modified BIC. because the methods estimate quantile regression functions, they possess an inherent robustness to extreme observations in the response values. We investigate the performance of prediction intervals based on automatic quantile regression splines and find that the loss of efficiency of this procedure is minimal in the normal linear homoscedastic model. In heteroscedastic linear models, it outperforms the classical normal theory prediction interval. A data example is provided to illustrate the use of the proposed methods.