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A constructive approach to the estimation of dimension reduction directions. (English) Zbl 1360.62196
Summary: 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., R. D. Cook and S. Weisberg, J. Am. Stat. Assoc. 86, No. 414, 328–332 (1991; Zbl 1353.62037); K.-C. Li, ibid. 86, No. 414, 316–327 (1991; Zbl 0742.62044); ibid. 87, No. 420, 1025–1039 (1992; Zbl 0765.62003)], 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., W. Härdle and T. M. Stoker, J. Am. Stat. Assoc. 84, No. 408, 986–995 (1989; Zbl 0703.62052); M. Hristache et al., Ann. Stat. 29, No. 6, 1537–1566 (2001; Zbl 1043.62052), Y. Xia et al., J. R. Stat. Soc., Ser. B, Stat. Methodol. 64, No. 3, 363–410 (2002; Zbl 1091.62028)], 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.

MSC:
62G08 Nonparametric regression and quantile regression
62G09 Nonparametric statistical resampling methods
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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