Tutorial: Dimension reduction in regression with a notion of sufficiency

• DOI: 10.5351/csam.2016.23.2.093
• Publication Date: 2016-04-04T00:25:18Z
• Authors: Jae Keun Yoo
• Keywords: Sufficient dimension reduction; Sliced inverse regression; Conditional probability distribution

In the paper, we discuss dimension reduction of predictors <TEX>${\mathbf{X}}{\in}{{\mathbb{R}}^p}$</TEX> in a regression <TEX>$Y{\mid}{\mathbf{X}}$</TEX> with notion sufficiency that is called sufficient reduction. reduction, original <TEX>${\mathbf{X}}$</TEX> are replaced by its lower-dimensional linear projection without loss information on selected aspects conditional distribution. Depending aspects, central subspace, mean subspace and <TEX>$k^{th}$</TEX>-moment defined investigated as primary interests. Then relationships among three subspaces changes for non-singular transformation studied. We two conditions to guarantee existence constrain marginal distribution <TEX>$Y{\mid}{\mathbf{X}}$</TEX>. A general approach estimate them also introduced along an explanation commonly assumed most methodologies.