An Asymptotic Theory for Sliced Inverse Regression

Abstract

Sliced inverse regression [Li (1989), (1991) and Duan and Li (1991)] is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, extremely easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. If $Y$ is the response and $X \in \mathbf{R}^p$ is the predictor, in order to implement sliced inverse regression, one requires an estimate of $\Lambda = E\{\operatorname{cov}(X\mid Y)\} = \operatorname{cov}(X) - \operatorname{cov}\{E(X\mid Y)\}$. The inverse regression of $X$ on $Y$ is clearly seen in $\Lambda$. One such estimate is Li's (1991) two-slice estimate, defined as follows: The data are sorted on $Y$, then grouped into sets of size 2, the covariance of $X$ is estimated within each group and these estimates are averaged. In this paper, we consider the asymptotic properties of the two-slice method, obtaining simple conditions for $n^{1/2}$-convergence and asymptotic normality. A key step in the proof of asymptotic normality is a central limit theorem for sums of conditionally independent random variables. We also study the asymptotic distribution of Greenwood's statistics in nonuniform cases.