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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)

62G07Density estimation
62E20Asymptotic distribution theory in statistics
62G20Nonparametric asymptotic efficiency
62G05Nonparametric estimation