Sliced inverse regression is a nonparametric method for achieving dimension reduction in regression problems. It is assumed that the conditional distribution of response given predictors depends only on linear combinations of . A key step in estimating the coefficients in the linear combinations is to estimate the expectation of the conditional covariance of given .
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- convergence and asymptotic normality of the two-slice estimator. To show that the assumption of the finiteness of the fourth moment of 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.