Penalized high‐dimensional M‐quantile regression: From L1 to Lp optimization

Abstract

AbstractQuantiles and expectiles have been receiving much attention in many areas such as economics, ecology, and finance. By means of Lp optimization, both quantiles and expectiles can be embedded in a more general class of M‐quantiles. Inspired by this point of view, we propose a generalized regression called Lp‐quantile regression to study the whole conditional distribution of a response variable given predictors in a heterogeneous regression setting. In this article, we focus on the variable selection aspect of high‐dimensional penalized Lp‐quantile regression, which provides a flexible application and makes a complement to penalized quantile and expectile regressions. This generalized penalized Lp‐quantile regression steers an advantageous middle course between ordinary penalized quantile and expectile regressions without sacrificing their virtues too much when 1 < p < 2, that is, offers versatility and flexibility with these ‘quantile‐like’ and robustness properties. We develop the penalized Lp‐quantile regression with scad and adaptive lasso penalties. With properly chosen tuning parameters, we show that the proposed estimators display oracle properties. Numerical studies and real data analysis demonstrate the competitive performance of the proposed penalized Lp‐quantile regression when 1 < p < 2, and they combine the robustness properties of quantile regression with the efficiency of penalized expectile regression. These properties would be helpful for practitioners.