AN ASYMPTOTIC THEORY FOR SLICED INVERSE REGRESSION

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 is-an-element-of R(p) is the predictor, in order to implement sliced inverse regression, one requires an estimate of LAMBDA = E{cov(X\Y)} = cov(X) - cov{E(X\Y)}. The inverse regression of X on Y is clearly seen in A. 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 n1/2-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.