Direct nonparametric conditional quantile estimation
Nonparametric conditional cumulative distribution function (CDF) estimation has emerged as a powerful tool having widespread potential application, which has led to a literature on indirect estimators of conditional quantile functions that are obtained via inversion of the nonparametrically estimated conditional CDF. Indirect nonparametric estimators of conditional quantiles that are based on an alternative characterisation of the quantile (i.e., as the function that minimises the expectation of the check function) have also appeared in the literature. In this paper, we propose a novel direct nonparametric approach that avoids indirect estimation of the quantile and is instead obtained via local averaging of a quantile kernel function itself. Relative to its indirect peers, the direct approach has a simple closed-form solution and is seen to be more efficient in nite-sample settings, particularly in the tail regions (our theoretical results underscore this property). Theoretical underpinnings are developed, data-driven smoothing parameter selection is provided, and Monte Carlo simulations and an empirical example are considered. The empirical example illustrates how the proposed approach can deliver more reasonable quantile and quantile derivative estimates than its indirect counterparts, particularly in tail regions.