Design-adaptive nonparametric estimation of conditional quantile derivatives

This paper proposes a new approach to constructing nonparametric estimators of conditional quantile functions and their derivatives with respect to conditioning variables. The new approach is intended specifically to produce estimators with biases that do not depend on the design density. This is in marked contrast to more conventional nonparametric estimators based on locally polynomial quantile regressions, the biases of which are characterised by asymptotic expansions in which the design density appears, at least at some order of approximation. The specific approach taken in this paper involves the kernel smoothing of the ratio of a preliminary nonparametric estimate of the conditional quantile function to another preliminary nonparametric estimate of the design density. Monte Carlo evidence indicates that the proposed estimators compare favourably to nonparametric estimators based on local polynomials. An empirical example exploring the relationship between individual earnings and age is also included. Additional technical details are contained in supplementary material available online.