Dimension reduction via marginal high moments in regression

Yin and Cook [2002. Dimension reduction for the conditional k-th moment in regression. J. Roy. Statist. Soc. B 64, 159-175] established a general equivalence between sliced inverse regression (sir) and a marginal moment method called Covk. In this note, we form a new marginal method called phdk and establish a general equivalence between sliced average variance estimation save, and Covk and phdk. We also show that in the population save is the most comprehensive method among all dimension reduction methods using the first two inverse moments. However, no similar relation was found for dimension reduction methods based on third inverse moments.