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Asymptotics of sliced inverse regression. (English) Zbl 0824.62036
Summary: Sliced Inverse Regression is a method for reducing the dimension of the explanatory variables \(x\) in nonparametric regression problems. K.- C. Li [J. Am. Stat. Assoc. 86, No. 414, 316-342 (1991; Zbl 0742.62044)] discussed a version of this method which begins with a partition of the range of \(y\) into slices so that the conditional covariance matrix of \(x\) given \(y\) can be estimated by the sample covariance matrix within each slice. After that the mean of the conditional covariance matrix is estimated by averaging the sample covariance matrices over all slices. T. Hsing and R. J. Carroll [Ann. Stat. 20, No. 2, 1040-1061 (1992; Zbl 0821.62019)] have derived the asymptotic properties of this procedure for the special case where each slice contains only two observations.
We consider the case that each slice contains an arbitrary but fixed number of \(y_ i\) and more generally the case when the number of \(y_ i\) per slice goes to infinity. The asymptotic properties of the associated eigenvalues and eigenvectors are also obtained.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62J99 Linear inference, regression
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