Weighted Quantile Regression Theory And Its Application

Abstract

As a robust data analysis technique, quantile regression has attracted extensive interest. In this study, the weighted quantile regression (WQR) technique is developed based on sparsity function. We first consider the linear regression model and show that the relative efficiency of WQR compared with least squares (LS) and composite quantile regression (CQR) is greater than 70% regardless of the error distributions. To make the pro- posed method practically more useful, we consider two nontrivial extensions. The first concerns with a nonparametric model. Local WQR estimate is introduced to explore the nonlinear data structure and shown to be much more efficient compared to other estimates under various non-normal error distributions. The second extension concerns with a multivariate problem where variable selection is needed along with regulation. We couple the WQR with penalization and show that under mild conditions, the penalized WQR en- joys the oracle property. The WQR has an intuitive formulation and can be easily implemented. Simulation is conducted to examine its finite sample performance and compare against alternatives. Analysis of mammal dataset is also conducted. Numerical studies are consistent with the theoretical findings and indicate the usefulness of WQR