A note on moment-based sufficient dimension reduction estimators

The two main groups of moment-based sufficient dimension reduction methods are the estimators for the central space and the estimators for the central mean space. The former group includes methods such as sliced inverse regression, sliced average variance estimation and sliced average third-moment estimation, while ordinary least squares and principal Hessian directions belong to the latter group. We provide unified frameworks for each group of estimators in this short note. The central space estimators can be unified as inverse conditional cumulants, while Stein’s Lemma is used to motivate the central mean space estimators.