Sliced Inverse Regression for Dimension Reduction: Comment

Pooled marginal slicing approach via SIR α with discrete covariables

“…The SIR-I estimates based on the first moment E(X|Y ) have been studied extensively; see for instance Duan and Li (1991), Li (1991), Carroll and Li (1992), Hsing and Carroll (1992), Zhu and Ng (1995), Kötter (1996), Saracco (1997Saracco ( , 1999, Aragon and Saracco (1997). But this approach is "blind" for symmetric dependencies (see Cook and Weisberg (1991) or Kötter (2000)). Then, SIR-II estimates based on the inverse conditional second moment V(X|Y ) have been suggested; see for instance Li (1991), Cook and Weisberg (1991) or Kötter (2000).…”

“…But this approach is "blind" for symmetric dependencies (see Cook and Weisberg (1991) or Kötter (2000)). Then, SIR-II estimates based on the inverse conditional second moment V(X|Y ) have been suggested; see for instance Li (1991), Cook and Weisberg (1991) or Kötter (2000). Hence these two approaches concentrate on the use of the inverse conditional moments E(X|Y ) or V(X|Y ) to find the e.d.r.…”

Sliced Inverse Regression

“…In other words, an estimate of S E(x|y) would provide a good approximation to S y|x . If condition (3) is not satisfied, then, following the suggestion of Li (1991) and Cook and Weisberg (1991), we can explore higher conditional moments of x given y or consider the socalled "central kth-moment subspace" (CKMS) (Yin and Cook 2002). We assume that S E(x|y) = S y|x and restrict our attention to SIR.…”

“…In contrast, the popular sequential test method can be easily used when the limit distribution of the estimator of the SIR matrix is available and its limiting variance is easy to estimate; otherwise, we must use a Monte Carlo approximation to simulate its distribution. Permutation (e.g., Cook and Weisberg 1991) or the bootstrap may be applicable if the consistency of these approximations can be verified.…”

“…Li (1991) derived the asymptotic results under the assumption that x is normally distributed, Hsing and Carroll (1992) proved the asymptotic normality of the slicing estimator when each slice contains two points, Zhu and Ng (1995) obtained the asymptotic normality for general cases, and Zhu and Fang (1996) studied the asymptotic behavior of a kernel estimator. Other important estimation approaches have also been treated in the literature, including sliced average variance estimates (SAVE) (Cook and Weisberg 1991;Cook 2000), parametric inverse regression (PIR) (Bura and Cook 2001a,b), and the proposed hybrid methods of Zhu, Ohtaki, and Li (2006) that are convex combinations of SIR and SAVE. Another relevant work is that of Cook and Li (2002).…”