DISCUSSION OF "BREAKDOWN AND GROUPS" BY P. L. DAVIES AND U. GATHER

1. Introductory remarks. It is a great pleasure for me to be invited to comment upon the nice and elegant and in parts thought-provoking paper by Davies and Gather. The authors also asked me specifically to comment upon the historical roots of the breakdown point (BP), and my thoughts about it. I shall try to do so, stressing in particular aspects and work that are not published. 2. Some thoughts with the definition of the breakdown point. In my thesis [Hampel (1968)] I developed what was later also called the “infinitesimal approach to robustness,” based on one-step Taylor expansions of statistics viewed as functionals (the “influence curves” or “influence functions”), a technology which for ordinary functions has long been indispensable in engineering and the physical sciences, and also for much theoretical work. However, it was always clear to me that this technology needed to be supplemented by an indication up to what distance (from the model distribution around which the expansion takes place) the linear expansions would be numerically, or at least semiquantitatively, useful. The simplest idea that came to my mind (simplicity being a virtue, also in view of Ockham’s razor) was the distance of the nearest pole of the functional (if it was unbounded); see the graphs in Hampel, Ronchetti, Rousseeuw and Stahel [(1986), pages 42, 48, 177]. Thus, right after defining the “bias function” (without using this term) as the (more complicated) bridge between model and pole, I introduced the “break-down point” on page 27 (Chapter C.4) of my thesis and, in a slight variant (by not requiring qualitative robustness anymore and therefore treating it as a purely global concept), as “breakdown point” on page 1894 in Hampel (1971). I was, of course, clearly inspired by Hodges (1967), whose intuition went in a similar direction, and by his “tolerance of extreme values”; however, his concept is not only much more limited, it is formally not even a special case of the breakdown point. [And contrary to