Group penalized expectile regression

The asymmetric least squares regression (or expectile regression) allows estimating unknown expectiles of the conditional distribution of a response variable as a function of a set of predictors and can handle heteroscedasticity issues. High dimensional data, such as omics data, are error prone and usually display heterogeneity. Such heterogeneity is often of scientific interest. In this work, we propose the Group Penalized Expectile Regression (GPER) approach, under high dimensional settings. GPER considers implementation of sparse expectile regression with group Lasso penalty and the group non-convex penalties. However, GPER may fail to tell which groups variables are important for the conditional mean and which groups of variables are important for the conditional scale/variance. To that end, we further propose a COupled Group Penalized Expectile Regression (COGPER) regression which can be efficiently solved by an algorithm similar to that for solving GPER. We establish theoretical properties of the proposed approaches. In particular, GPER and COGPER using the SCAD penalty or MCP is shown to consistently identify the two important subsets for the mean and scale simultaneously. We demonstrate the empirical performance of GPER and COGPER by simulated and real data.