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Kernel dimension reduction in regression. (English) Zbl 1168.62049
Summary: We present a new methodology for sufficient dimension reduction (SDR). Our methodology derives directly from the formulation of SDR in terms of the conditional independence of the covariate $X$ from the response $Y$, given the projection of $X$ on the central subspace [cf. {\it K.-C. Li}, J. Am. Stat. Assoc. 86, No. 414, 316--342 (1991; Zbl 0742.62044); and “Regression graphics. Ideas for studying regressions through graphics.” New York: Wiley (1998; Zbl 0903.62001)]. We show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel Hilbert spaces and we show how this characterization leads to an $M$-estimator for the central subspace. The resulting estimator is shown to be consistent under weak conditions; in particular, we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for SDR methods. We also present empirical results showing that the new methodology is competitive in practice.

##### MSC:
 62H05 Characterization and structure theory (Multivariate analysis) 46N30 Applications of functional analysis in probability theory and statistics 62J99 Linear statistical inference 62H99 Multivariate analysis 62J02 General nonlinear regression 65C60 Computational problems in statistics
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