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

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