A general framework to extend sufficient dimension reductions to the cases of the mixture multivariate elliptical distributions

Abstract

In the sufficient dimension reduction (SDR), many methods depend on some assumptions on the distribution of predictor vector, such as the linear design condition (L.D.C.), the assumption of constant conditional variance, and so on. The mixture distributions emerge frequently in practice, but they may not satisfy the above assumptions. In this article, a general framework is proposed to extend various SDR methods to the cases where the predictor vector follows the mixture elliptical distributions, together with the asymptotic property for the consistency of the kernel matrix estimators. For illustration, the extensions of several classical SDR approaches under the proposed framework are detailed. Moreover, a method to estimate the structural dimension is given, together with a procedure to check an assumption called homogeneity. The proposed methodology is illustrated by simulated and real examples.