Estimating the Central Kth Moment Space via an Extension of Ordinary Least Squares

Various sufficient dimension reduction methods have been proposed to find linear combinations of predictor X, which contain all the regression information of Y versus X. If we are only interested in the partial information contained in the mean function or the kth moment function of Y given X, estimation of the central mean space (CMS) or the central kth moment space (CKMS) becomes our focus. However, existing OLS-type estimators for CMS and CKMS require a linearity assumption on the predictor distribution. In this paper, we relax this stringent limitation via the notion of central solution space (CSS). Central kth moment solution space is introduced and its estimators are compared with existing methods by simulation.