Efficient estimation in expectile regression using envelope models

Abstract

As a generalization of the classical linear regression, expectile regression (ER) explores the relationship between the conditional expectile of a response variable and a set of predictor variables. ER with respect to different expectile levels can provide a comprehensive picture of the conditional distribution of the response variable given the predictors. We adopt an efficient estimation method called the envelope model ([8]) in ER, and construct a novel envelope expectile regression (EER) model. Estimation of the EER parameters can be performed using the generalized method of moments (GMM). We establish the consistency and derive the asymptotic distribution of the EER estimators. In addition, we show that the EER estimators are asymptotically more efficient than the ER estimators. Numerical experiments and real data examples are provided to demonstrate the efficiency gains attained by EER compared to ER, and the efficiency gains can further lead to improvements in prediction.