A constructive approach to the estimation of dimension reduction directions

In this paper we propose two new methods to estimate the dimension-reduction directions of the central subspace (CS) by constructing a regression model such that the directions are all captured in the regression mean. Compared with the inverse regression estimation methods [e.g., J. Amer Statist. Assoc. 86 (1991) 328-332, J Amer Statist. Assoc. 86 (1991) 316-342, J Amer Statist. Assoc. 87 (1992) 1025-1039], the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable, and have better perforrnance than the inverse regression estimation methods for finite samples. Compared with the direct regression estimation methods [e.g., J. Amer. Statist. Assoc. 84 (1989) 986-995, Ann. Statist. 29 (2001) 1537-1566, J R. Stat. Soc. Ser B Stat. Methodol. 64 (2002) 363-410], which can only estimate the directions of CS in the regression mean, the new methods can detect the directions of CS exhaustively. Consistency of the estimators and the convergence of corresponding algorithms are proved.