Sufficient dimension reduction with mixture multivariate skew-elliptical distributions

In inverse regression-based methodologies for sufficient dimension reduction, the ellipticity (or slightly more generally, the linearity condition) of the predictor vector, and constant conditional variance assumption are widely used conditions of which researchers however concern its restrictiveness. In this project, Stein Lemma is systematically generalized to the class of mixture multivariate skew-elliptical distributions in different scenarios and to identify and estimate the central subspace. Within this class, necessary and sufficient conditions are explored for ordinary least squares type method (OLS) and principal Hessian directions method (pHd) for identifying the central and central mean subspace. It may also provide a way to do adjustments such that the central subspace can still be identifiable when OLS and pHd fail to work. Further, we also explore the potential of sliced inverse regression (SIR) and sliced average variance estimation (SAVE) under the mixture multivariate skew- elliptical distribution scenarios.

Clearly, this research project investigates some important problems in sufficient dimension reduction, and expect the new methodologies and theories developed in the course of the project will have a lasting impact in this field.