Using central k-th moment subspace, Yin and Cook (2002) decomposed Sliced Inverse Regression (SIR) into focused marginal moment method, COV_k. Under central subspace, Ye and Weiss (2003) also studied general relations among SIR, sliced average variance estimate (SAVE), and Principal Hessian Direction (PHD) to form a new class of dimension reduction methods. In this note, by forming a new marginal method via inverse second moment called PHD_k for estimating directions in the central k-th moment subspace, we are able to decompose SAVE into some focused marginal moment methods. Among all the methods using first and second inverse moments we show that in theory SAVE is the most comprehensive method. However, we find that such kind of relations do not hold for methods using third inverse moment.
Dimension Reduction Via Marginal High Moments in Regression