High-Dimensional Heteroscedastic Regression

• Activity Number: 369
• Type: Contributed
• Date/Time: Tuesday, August 11, 2015 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Yuwen Gu* and Hui Zou
• Companies: University of Minnesota and University of Minnesota
• Keywords: High dimensions ; Heteroscedasticity ; Expectile ; LLA

Abstract

In high-dimensional data, heteroscedasticity or non-constant error variance is commonly present but often receives little attention. In statistical problems where the driving explanatory variables for the variance are relevant, it is desired that variable selection and parameter estimation can be done simultaneously for both the mean and the variance. For a general class of heteroscedastic models in which different segments of the conditional distribution of the response variable given all the explanatory variables may depend on different sets of relevant variables, whose sizes are assumed to be small, we propose the sparse expectile regression as a way to detect heteroscedasticity. The expectile regression, which is a variant of OLS, employs an asymmetric squared error loss in a similar way as the check loss for quantile regression, and thus can model the entire conditional distribution in a regression model. We show that when regularized with folded concave penalties, the sparse expectile regression enjoys the strong oracle property and the local linear approximation (LLA) algorithm can be used to find the oracle solution with overwhelming probability.