Tutorial: Dimension reduction in regression with a notion of sufficiency

Title & Authors
Tutorial: Dimension reduction in regression with a notion of sufficiency
Yoo, Jae Keun;

Abstract
In the paper, we discuss dimension reduction of predictors $\small{{\mathbf{X}}{\in}{{\mathbb{R}}^p}}$ in a regression of $\small{Y{\mid}{\mathbf{X}}}$ with a notion of sufficiency that is called sufficient dimension reduction. In sufficient dimension reduction, the original predictors $\small{{\mathbf{X}}}$ 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 $\small{k^{th}}$-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 $\small{{\mathbf{X}}}$ are studied. We discuss the two conditions to guarantee the existence of the three subspaces that constrain the marginal distribution of $\small{{\mathbf{X}}}$ and the conditional distribution of $\small{Y{\mid}{\mathbf{X}}}$. A general approach to estimate them is also introduced along with an explanation for conditions commonly assumed in most sufficient dimension reduction methodologies.
Keywords
central subspace;central $\small{k^{th}}$-moment subspace;central mean subspace;dimension reduction subspace;regression;sufficient dimension reduction;
Language
English
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