Group LASSO for High-Dimensional Sparse Quantile Regression Models

• Activity Number: 83
• Type: Contributed
• Date/Time: Sunday, July 31, 2011 : 4:00 PM to 5:50 PM
• Sponsor: IMS
• Author(s): Kengo Kato*+
• Companies: Hiroshima University
• Keywords: additive model ; group Lasso ; non-asymptotic bound ; quantile regression

Abstract

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 ``active'' variables 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. 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 primitive 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.