On the stability of bootstrap estimators

TL;DR

This paper demonstrates that bootstrap methods for estimators derived from continuous operators on probability measures are stable in terms of robustness, with applications to support vector machines using shifted loss functions.

Contribution

It establishes the stability of bootstrap approximations for a broad class of estimators, including support vector machines, under qualitative robustness.

Findings

Bootstrap estimators are stable for continuous operators on probability measures.

Support vector machines with shifted loss functions are included as special cases.

The results ensure robustness of bootstrap methods in these settings.

Abstract

It is shown that bootstrap approximations of an estimator which is based on a continuous operator from the set of Borel probability measures defined on a compact metric space into a complete separable metric space is stable in the sense of qualitative robustness. Support vector machines based on shifted loss functions are treated as special cases.