Convergence rate for nonparametric quantile regression with a total variation penalty

Quantile regression with a total variation penalty was previously proposed due to its computational expediency as well as its local adaptiveness. However, the convergence rate of the method in this setting has been not rigorously established. In this short communication, we establish the convergence rate of O-p(n(-1/3)) for the penalized estimator which is the same as in penalized least squares regression. Different from penalized least squares regression, in order to deal with the quantile loss function, we heavily rely on the Rademacher complexity of the class of functions of bounded variation.