Discussion of “ Breakdown and Groups ”

1. General comments. The interesting paper of Davies and Gather (henceforth [DG]) pulls together results on upper bounds on the breakdown value of translation equivariant location estimators [Donoho (1982)], regression esti-mators [Rousseeuw (1984)] and affine equivariant scatter estimators [Davies (1987)] into a single framework of group equivariance. I can only agree with them on the important role of the latter notion in obtaining nontrivial bounds. [By the way, I prefer the term breakdown value myself because it is not a point, and the term " value " captures both its dimension (one) and its orientation (we aim for higher, not lower values).] The theory in [DG] is formulated for estimators that are uniquely defined, but it seems to work just as well in the general case. Then T (P) is a set, and we can follow the implicit convention of saying that it breaks down when any member of T (P) does. We only need to redefine D(T (P), T (Q)) in (2.4) as a supremum over all pairs of members of T (P) and T (Q). The new applications of the theory are fascinating, for example, to the Michaelis–Menten model (with a nontrivial bound) and logistic regression (without one). I am less convinced by the fragility argument illustrated by the difference between the contaminated samples (6.2) and (6.3). It is true that this proof of the upper bound only covers (6.2), but in some sense that is enough since breakdown is a worst-case concept and the bound is not specific for the median but for all translation estimators. But anyway, the behavior of the median at (6.3) can be derived from that at (6.2) by a variety of other properties that it possesses. For instance, its monotonicity property alone suffices. Or we can use the property that when you move the observations over distances of at most δ, the median changes by at most δ. This holds for any δ > 0, and is a Lipschitz property for the metric on samples defined as