Nonparametric quantile regression with multivariate covariates
is a difficult estimation. To reduce the dimensionality while still retaining the
flexibility of nonparametric model, the single-index regression is often used to model
the conditional quantile of a response variable. In this paper, we focus on the variable
selection aspect of single-index quantile regression. Based on the minimized average
loss estimation (MALE), the variable selection is done by minimizing the average loss
with SCAD penalty. Under some mild conditions, we demonstrate the oracle properties
about SCAD variable section of single-index quantile regression. Furthermore, the
algorithm of the variable selection of SCAD penalized quantile regression is given.
Some simulations are done to illustrate the performance of the proposed methods.