Generalized kernel-based inverse regression methods for sufficient dimension reduction

The linearity condition and the constant conditional variance assumption popularly used in sufficient dimension reduction are respectively close to elliptical symmetry and normality. However, it is always the concern about their restrictiveness. In this article, we give systematic studies to provide insight into the reasons why the popularly used sliced inverse regression and sliced average variance estimation need these conditions. Then we propose a new framework to relax these conditions and suggest generalized kernel-based inverse regression methods to handle a class of mixture multivariate unified skew-elliptical distributions.