Asymptotics for kernel estimate of sliced inverse regression

Abstract

To explore nonlinear structures hidden in high-dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression, Li and Duan proposed the sliced inverse regres-Ž . sion SIR method which is simple to use. In this paper, the asymptotic properties of the kernel estimate of sliced inverse regression are investigated. It turns out that regardless of the kernel function, the asymptotic distribution remains the same for a wide range of smoothing parameters.

Sliced Inverse Regression

“…Condition (B) assumes the smoothness of the inverse regression m(y). It is slightly stronger than that of Zhu and Ng (1995) when p is fixed; therefore, it is quite mild. As for conditions (C) and (D), they are special for our problem.…”

“…A similar definition of total variation has been given by Hsing and Carroll (1992) and Zhu and Ng (1995), except for the value √ p in the denominator. We have this value because in the Euclidean norm of m(y * (i+1) ) − m(y * (i) ) there are p terms to be summed.…”

“…As for conditions (C) and (D), they are special for our problem. In the case where p is fixed and the distribution F n of y i is also independent of n, conditions (C) and (D) are reduced to those of Zhu and Ng (1995). Hence the conditions are needed to uniformly bind the distribution series F n (·).…”

“…The choice of the number, c, of data points in each slice is of practical importance. When p is fixed, Zhu and Ng (1995) derived that for a wide range of c, the convergence of the estimate can be achieved. When p varies with n, we have not obtained a similar result for the case with c → ∞ and c/n → 0 as n → ∞ due to technical difficulties.…”

“…This method is popular in the literature. 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.…”