In regression analysis, it is difficult to uncover the dependence
relationship between a response variable and a covariate vector
when the dimension of the covariate vector is high. To reduce the
dimension of the covariate vector, one approach is sufficient
dimension reduction. Sufficient dimension reduction is based on
the assumption that the response variable relates to only a few
linear combinations of the covariate vector. Thus, by replacing
the covariate vector with these linear combinations, sufficient
dimension reduction achieves dimension reduction. The goal of
sufficient dimension reduction is to estimate the space spanned
by these linear combinations of the covariate vector. We denote
this space by S.
In this thesis, we give an introductory review on three important
sufficient dimension reduction methods. They are Sliced Inverse
Regression (SIR), Sliced Average Variance Estimate (SAVE) and
Principle Hessian Directions (pHd). Li proposed SIR in 1991. SIR
is a method that exploits the simplicity of the inverse
regression. Given the univariate response variable and the high
dimensional covariate, it is much easier to regress the covariate
against the response variable than the other way around.
Motivated by a theorem that connects forward regression and
inverse regression, SIR estimates S using inverse regression
lines. Since SIR uses first moments only, it fails when there
exists symmetry dependence between the response variable and the
covariate. To make up for this defect, Cook proposed SAVE in a
comment on SIR in 1991. SAVE follows the general lines of SIR but
uses second moments as well as first moments to estimate S. pHd
is also a second moment method. Li developed pHd in 1992 based on
the observation that the eigenvectors for the Hessian matrices of
the regression function are closely related to the basis vectors
of S. Therefore pHd provides an estimate of S by using these
eigenvectors.
To compare these methods, a simulation study is presented at the
end. From the simulation results, SIR is the most efficient
method and SAVE is the most time consuming method. Since SIR
fails when symmetry dependence exists, we recommend pHd when
symmetry dependence presents and SIR in other cases
