There is a vast literature on robust estimators, but in some situations it is still not easy to make inferences, such as confidence regions and hypothesis testing. This is
mainly due to the following facts. On one hand, in most situations, it is difficult to derive the exact distribution of the estimator. On the other one, even if its asymptotic behaviour is known, in many cases, the convergence to the limiting distribution may be rather slow, so bootstrap methods are preferable since they often give better small sample results. However, resampling methods have several disadvantages including the propagation of anomalous data all along the new samples. In this paper, we discuss the problems arising in the bootstrap when outlying observations are present. We argue that it is preferable to use a robust bootstrap rather than to bootstrap robust estimators and we discuss a robust bootstrap method, the Influence Function Bootstrap denoted IFB. We illustrate the performance of the IFB intervals in the univariate location case and in the logistic regression model. We derive some asymptotic properties of the IFB. Finally, we introduce a generalization of the Influence Function Bootstrap in order to improve the IFB behaviour.

CEMAT - Center for Computational and Stochastic Mathematics