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An asymptotic theory for sliced inverse regression. (English) Zbl 0821.62019

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)

##### MSC:
 62G07 Density estimation 62E20 Asymptotic distribution theory in statistics 62G20 Nonparametric asymptotic efficiency 62G05 Nonparametric estimation