Consistency and robustness of kernel-based regression in convex risk minimization

Details Consistency and robustness of kernel-based regression in convex risk minimization Consistency and robustness of kernel-based regression in convex risk minimization

2007-09-05

arxiv.org/abs/0709.0626v1

Bernoulli 2007, Vol. 13, No. 3, 799-819

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.3150/07-BEJ5102 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statist

Scientific paper

10.3150/07-BEJ5102

We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function $L$ of the KBR method and the tail of the response variable. We then establish the $L$-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to ``learn''. Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical $M$ estimators is discussed.

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