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Dimension reduction for nonelliptically distributed predictors. (English) Zbl 1160.62050
Summary: Sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor, or, when such conditions are not satisfied, rely on marginal transformations or reweighting to fulfill them approximately. For example, a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distributions. We reformulate the commonly used dimension reduction methods, via the notion of “central solution space” to circumvent the requirements of such strong assumptions, while at the same time preserve the desirable properties of the classical methods, such as \(\sqrt n\)-consistency and asymptotic normality. Imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the “curse of dimensionality”, but the developments of this paper show that this need not be the case. The new methods will be compared with existing methods by simulation and applied to a data set.

62H12 Estimation in multivariate analysis
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G09 Nonparametric statistical resampling methods
62E20 Asymptotic distribution theory in statistics
65C60 Computational problems in statistics (MSC2010)
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