Local asymptotics for quantile smoothing splines

Abstract

Quantile smoothing splines were introduced by Koenker, Ng and Portnoy as natural and appealing estimates of conditional quantiles of response variables. The natural setting for the problem considers minimization of a weighted combination of a "fit" penalty and a "roughness" penalty over the space of functions whose derivatives have bounded variation. Although this space is not traditional, Shen has shown recently that the quantile smoothing splines do indeed converge at the usual optimal rate $n^{-2/5)$ in various norms. Here, local asymptotic results are obtained by establishing Bahadur representations for local parameters of the splines. These are used to obtain local rates of convergence, to establish uniform convergence rates, to provide local distribution theory for quantile B-splines and to expand the "fit" measure in order to analyze an information criterion for determining the smoothing parameter. Examples of using derivatives of the smoothing splines for estimating jump functions are also presented.