Local weighted composite quantile estimation and smoothing parameter selection for nonparametric derivative function

Estimating derivatives is of primary interest as it quantitatively measures the rate of change of the relationship between response and explanatory variables. We propose a local weighted composite quantile method to estimate the gradient of an unknown regression function. Because of the use of weights, a data-driven weighting scheme is discussed for maximizing the asymptotic efficiency of the estimators. We derive the leading bias, variance and normality of the estimator proposed. The asymptotic relative efficiency is investigated and reveals that the new approach provides a highly efficient alternative to the local least squares, the local quantile regression and the local composite quantile regression methods. In addition, a fully automatic bandwidth selection method is considered and is shown to deliver the bandwidth with oracle property meaning that it is asymptotically equivalent to the optimal bandwidth if the true gradient were known. Simulations are conducted to compare different estimators and an example is used to illustrate their performance. Both simulation and empirical results are consistent with our theoretical findings.