Confidence intervals for nonparametric quantile regression: an emphasis on smoothing splines approach

In this paper we consider the problem of constructing confidence intervals for nonparametric quantile regression with an emphasis on smoothing splines. The mean-based approaches for smoothing splines of Wahba (1983) and Nychka (1988) may not be efficient for constructing confidence intervals for the underlying function when the observed data are non-Gaussian distributed, for instance if they are skewed or heavy-tailed. This paper proposes a method of constructing confidence intervals for the unknown th quantile function (0<<1) based on smoothing splines. In this paper we investigate the extent to which the proposed estimator provides the desired coverage probability. In addition, an improvement based on a local smoothing parameter that provides more uniform pointwise coverage is developed. The results from numerical studies including a simulation study and real data analysis demonstrate the promising empirical properties of the proposed approach.