A constructive approach to the estimation of dimension reduction directions
Yingcun Xia
Source: Ann. Statist. Volume 35, Number 6
(2007), 2654-2690.
Abstract
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 performance 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.
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Keywords: Conditional density function; convergence of algorithm; double-kernel smoothing; efficient dimension reduction; root-n consistency
Full-text: Open access
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1201012976
Digital Object Identifier: doi:10.1214/009053607000000352
Mathematical Reviews number (MathSciNet): MR2382662
Zentralblatt MATH identifier: 05241119
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