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An asymptotic theory for sliced inverse regression. (English) Zbl 0954.62531

Summary: Sliced Inverse Regression (S.I.R.) is a method for reducing the dimension of the explanatory variable x in nonparametric regression problems. K. C. Li [J. Am. Statist. Assoc. 86, 316-342 (1991)] considers a general regression model of the form

y=g(x ' β 1 ,,x ' β K ,ε)

with an arbitrary and unknown link function g, and studies a link-free and distribution-free method for estimating E, the space spanned by the β k ’s, called the effective dimension reduction (e.d.r.) space. It is widely applicable, easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. The method begins with a partition of the range of y into a fixed number of slices. Let us denote T(·) this partition. The conditional mean of x given T(y) is then estimated by the sample mean within each slice.

MSC:
62G05Nonparametric estimation
62J02General nonlinear regression