Sufficient dimension reduction with mixture multivariate skew-elliptical distributions

In inverse regression-based methodologies for sufficient dimension reduction, ellipticity (or slightly more generally, the linearity condition) of the predictor vector is a widely used condition, though there is concern over its restrictiveness. In this paper, Stein's Lemma is generalized to the class of mixture multivariate skewelliptical distributions in different scenarios to identify and estimate the central subspace. Within this class, necessary and sufficient conditions are explored for the simple covariance between the response (or its function) and the predictor vector to identify the central subspace. Further, we provides a way to do adjustments such that the central subspace can still be identifiable when this simple covariance fails to work. Simulations are used to assess the performance of the results and compare with existing methods. A data example is analysed for illustration.