Hsing, Tailen; Carroll, Raymond J. An asymptotic theory for sliced inverse regression. (English) Zbl 0821.62019 Ann. Stat. 20, No. 2, 1040-1061 (1992). Sliced inverse regression is a nonparametric method for achieving dimension reduction in regression problems. It is assumed that the conditional distribution of response \(Y\) given predictors \(X\) depends only on \(K\) linear combinations of \(X\). A key step in estimating the \(K\) coefficients in the linear combinations is to estimate the expectation of the conditional covariance of \(X\) given \(Y\).K.-C. Li [J. Am. Stat. Assoc. 86, No. 414, 316-342 (1991; Zbl 0742.62044)] suggested a two-slice estimator for this expectation. By developing a central limit theorem for the sum of conditionally independent random variables, the authors in this paper proved the root- \(n\) convergence and asymptotic normality of the two-slice estimator. To show that the assumption of the finiteness of the fourth moment of \(X\) in the major results of this paper is essentially necessary, the asymptotic distribution of Greenwood’s statistic [M. Greenwood, J. R. Stat. Soc., Ser. A 109, 85-110 (1946)] in nonuniform cases is also studied. Reviewer: D.Tu (Ottawa) Cited in 3 ReviewsCited in 73 Documents MSC: 62G07 Density estimation 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation Keywords:root-\(n\) convergence; sliced inverse regression; dimension reduction; conditional distribution; linear combinations; conditional covariance; two-slice estimator; central limit theorem; asymptotic normality; finiteness of the fourth moment; asymptotic distribution of Greenwood’s statistic Citations:Zbl 0742.62044 PDFBibTeX XMLCite \textit{T. Hsing} and \textit{R. J. Carroll}, Ann. Stat. 20, No. 2, 1040--1061 (1992; Zbl 0821.62019) Full Text: DOI