## Abstract

We propose the Bouligand influence function (BIF) as a

new concept for robust statistics. The BIF is a modification of F.R.

Hampel's influence function (IF) and is based on a special cone

derivative instead of the usual G{\^a}teaux-derivative. If the BIF

does exist, then the IF does also exist and both are equal. The

usefulness of Bouligand-derivatives to robust statistics is

explained.

In the second part of the talk we apply the BIF to support vector

machines based on a non-smooth loss function for which the influence

was unknown. We show for the regression case that many support

vector machines based on a Lipschitz continuous loss function and a

bounded kernel have a bounded BIF and hence also have a bounded IF.

In this respect such SVMs are therefore robust. Special cases are

SVMs based on the $\epsilon$-insensitive loss, Huber's loss, and

kernel based quantile regression based on the pinball loss.

new concept for robust statistics. The BIF is a modification of F.R.

Hampel's influence function (IF) and is based on a special cone

derivative instead of the usual G{\^a}teaux-derivative. If the BIF

does exist, then the IF does also exist and both are equal. The

usefulness of Bouligand-derivatives to robust statistics is

explained.

In the second part of the talk we apply the BIF to support vector

machines based on a non-smooth loss function for which the influence

was unknown. We show for the regression case that many support

vector machines based on a Lipschitz continuous loss function and a

bounded kernel have a bounded BIF and hence also have a bounded IF.

In this respect such SVMs are therefore robust. Special cases are

SVMs based on the $\epsilon$-insensitive loss, Huber's loss, and

kernel based quantile regression based on the pinball loss.

Original language | English |
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Title of host publication | 15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium) |

Publication status | Published - 18 Oct 2007 |

### Publication series

Name | 15th Annual Meeting of the Belgian Statistical Society, Oct 18-20, 2007, Antwerp (Belgium) |
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## Keywords

- SVM
- robustness
- Bouligand