Kernel dimension reduction in regression

Details Kernel dimension reduction in regression Kernel dimension reduction in regression

2009-08-13

arxiv.org/abs/0908.1854v1

Annals of Statistics 2009, Vol. 37, No. 4, 1871-1905

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-AOS637 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/08-AOS637

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. J. Amer. Statist. Assoc. 86 (1991) 316--342 and Regression Graphics (1998) Wiley]. 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.

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