Tutorial: Dimension reduction in regression with a notion of sufficiency Yoo, Jae Keun;
In the paper, we discuss dimension reduction of predictors in a regression of with a notion of sufficiency that is called sufficient dimension reduction. In sufficient dimension reduction, the original predictors are replaced by its lower-dimensional linear projection without loss of information on selected aspects of the conditional distribution. Depending on the aspects, the central subspace, the central mean subspace and the central -moment subspace are defined and investigated as primary interests. Then the relationships among the three subspaces and the changes in the three subspaces for non-singular transformation of are studied. We discuss the two conditions to guarantee the existence of the three subspaces that constrain the marginal distribution of and the conditional distribution of . A general approach to estimate them is also introduced along with an explanation for conditions commonly assumed in most sufficient dimension reduction methodologies.
central subspace;central -moment subspace;central mean subspace;dimension reduction subspace;regression;sufficient dimension reduction;