Kernel methods for estimating derivatives of conditional quantiles

A collection of quantile regression functions gives a picture of the conditional distribution of the response given the covariates. However, it cannot be used directly to make a firm conclusion on the effects of the covariates. The derivatives of conditional quantiles, instead, are of immediate use for this purpose. They measure how rapidly the conditional quantiles change as the covariates vary, not only in the center of the population, but also in its upper and lower tails. In this paper we consider estimation of the derivatives of conditional quantiles. The estimators suggested in this paper are based on the double-kernel approach of [Yu, K., & Jones, M. C. (1998). Local linear quantile regression. Journal of the American Statistical Association, 93, 228–237] and on the local logistic approach of [Lee, Y. K., Lee, E. R., & Park, B. U. (2006). Conditional quantile regression by local logistic regression. Journal of Nonparametric Statistics, 18, 357–373]. We derive the asymptotic distributions of the two estimators, and compare their finite sample performance via a simulation study.