Dimension reduction for the conditional kth 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 kth moment function of Y given X, estimation of the central mean space or the central kth moment space becomes our focus. However, existing estimators for the central mean space and the central kth moment space require a linearity assumption on the predictor distribution. In this paper, we relax this stringent assumption via the notion of central kth moment solution space. Simulation studies and analysis of the Massachusetts college data set confirm that our proposed estimators of the central kth moment space outperform existing methods for non-elliptically distributed predictors.