Summary: To explore nonlinear structures hidden in high-dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression, N. Duan
and K. C. Li
[ibid. 19, No. 2, 505-530 (1991; Zbl 0738.62070
)] proposed the sliced inverse regression (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.