Group Lasso for high dimensional sparse quantile regression models

By Kengo Kato. 2011

52 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8e81286f01dfb370e06597234408eb144657fb2e

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Citations in the corpus

Program evaluation and causal inference with high-dimensional data (2013, methodology): “For example, this type of orthogonality was used by Neyman (1979) in low-dimensional settings to deal with crudely estimated parametric nuisance parameters. See also Newey (1990), Andrews (1994b), Newey (1994), Robins and Rotnitzky (1995), and Linton (1996) for the use of this condition in semi-para”
Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems (2013, background): “3) attain (essentially) optimal rates √ (s log p)/n of convergence for estimating the entire median regression function, as has been shown in [28, 3, 15, 31] and in [3].”
Valid Post-Selection Inference in High-Dimensional Approximately Sparse Quantile Regression Models (2013, methodology): “The former can be achieved via 1-penalized quantile regression (Belloni andChernozhukov 2011; Kato 2011) or post-selection quantile regression based on 1-penalized quantile regression (Belloni and Chernozhukov 2011).”
High-Dimensional Methods and Inference on Structural and Treatment Effects (2013, background): “For example, van de Geer (2008) and Belloni, Chernozhukov, and Wei (2013) consider high-dimensional generalized linear regression, and Belloni and Chernozhukov (2011), Belloni, Chernozhukov, and Kato (2013), and Kato (2011) consider quantile regression.”
Program evaluation with high-dimensional data (2013, background): “The following maximal inequality is due to Chernozhukov, Chetverikov, and Kato (2012).”