Rejoinder to “ Breakdown and Groups

We thank all the discussants for their contributions and in particular we wish to thank Hampel. The concept of breakdown point goes back to his Ph.D. thesis [Hampel (1968)] and he was the first to exhibit a high breakdown equivariant regression estimate now known as the least median of squares [Hampel (1975)], a fact which is sometimes forgotten. These two sources are the starting point of the present discussion. In his contribution Hampel gives us insight into the thoughts which led to his definition of breakdown point, intended as it was to complement the infinitesimal behavior of a functional as described by the influence function. Hampel emphasizes that equivariance considerations were not part of his definition and he had in mind correlation statistics " where there is no equivariance at all. " He considers correlation in some detail and, as we disagree with him on this very topic, we give a detailed analysis of correlation statistics in our rejoinder. We hope that this will help clarify the issues involved. 1. On breakdown. The first signification of the word " breakdown " given in the Oxford Dictionary starts with the following subsignification: " 1. a. The act of breaking and falling down: a ruinous downfall, a collapse. " 2. Breakdown to points and variations. Genton and Lucas and Oja argue for the usefulness of the breakdown concept in situations not covered by the results of our paper. They claim that at least in an intuitive sense breakdown occurs if the value of a functional is driven to the boundary or to an interior point which is independent of the uncontaminated sample. A formal definition of breakdown point is given which is intended to cover such possibilities. A first version is to be found in Genton and Lucas (2003) and is referred to by Oja. It defines the breakdown point as the smallest amount of contamination which can cause the statistic to assume only a finite number