Abstract: We investigate the simultaneous estimation and inference of the central mean subspace and central variance subspace to reduce the effective number of covariates that predict, respectively, the mean and variability of the response variable. We study the estimation, inference and efficiency properties under different scenarios, and further propose a class of locally efficient estimators when the truly efficient estimator is not practically available. This partially explains the necessity of some dimension-reduction assumptions that are commonly imposed on the conditional mean function in estimating the central variance subspace. Comprehensive simulation studies and a data analysis are performed to demonstrate the finite sample performance and efficiency gain of the locally efficient estimators in comparison with existing estimation procedures.

Key words and phrases: Central mean subspace, central variance subspace, dimension reduction, location-scale family, semiparametric efficiency.