*(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.

##### MSC:

62G07 | Density estimation |

62E20 | Asymptotic distribution theory in statistics |

62G20 | Nonparametric asymptotic efficiency |

62G05 | Nonparametric estimation |