  This paper studies the statistical properties of the group Lasso estimator
for high dimensional sparse quantile regression models where the number of
explanatory variables (or the number of groups of explanatory variables) is
possibly much larger than the sample size while the number of variables in
"active" groups is sufficiently small. We establish a non-asymptotic bound on
the $\ell_{2}$-estimation error of the estimator. This bound explains
situations under which the group Lasso estimator is potentially
superior/inferior to the $\ell_{1}$-penalized quantile regression estimator in
terms of the estimation error. We also propose a data-dependent choice of the
tuning parameter to make the method more practical, by extending the original
proposal of Belloni and Chernozhukov (2011) for the $\ell_{1}$-penalized
quantile regression estimator. As an application, we analyze high dimensional
additive quantile regression models. We show that under a set of suitable
regularity conditions, the group Lasso estimator can attain the convergence
rate arbitrarily close to the oracle rate. Finally, we conduct simulations
experiments to examine our theoretical results.
