Dimension reduction for the conditional th moment via central solution space

Sufficient dimension reduction aims at finding transformations of predictor X without losing any regression information of Y versus X . If we are only interested in the information contained in the mean function or the k th moment function of Y given X , estimation of the central mean space or the central k th moment space becomes our focus. However, existing estimators for the central mean space and the central k th moment space require a linearity assumption on the predictor distribution. In this paper, we relax this stringent assumption via the notion of central k th moment solution space. Simulation studies and analysis of the Massachusetts college data set confirm that our proposed estimators of the central k th moment space outperform existing methods for non-elliptically distributed predictors.