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In this paper, we propose two new methods to estimate the dimension-reductiondirections of the central subspace (CS) by constructing a regression model suchthat the directions are all captured in the regression mean. Compared with theinverse regression estimation methods (e.g. Li, 1991, 1992; Cook and Weisberg,1991), the new methods require no strong assumptions on the design ofcovariates or the functional relation between regressors and the responsevariable, and have better performance than the inverse regression estimationmethods for finite samples. Compared with the direct regression estimationmethods (e.g. H\"ardle and Stoker, 1989; Hristache, Juditski, Polzehl andSpokoiny, 2001; Xia, Tong, Li and Zhu, 2002), which can only estimate thedirections of CS in the regression mean, the new methods can detect thedirections of CS exhaustively. Consistency of the estimators and theconvergence of corresponding algorithms are proved.

A constructive approach to the estimation of dimension reduction directions