  The concept of breakdown point was introduced by Hampel [Ph.D. dissertation
(1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896]
and developed further by, among others, Huber [Robust Statistics (1981). Wiley,
New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983)
157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context
of location, scale and regression problems. Attempts to extend the concept to
other situations have not met with general acceptance. In this paper we argue
that this is connected to the fact that in the location, scale and regression
problems the translation and affine groups give rise to a definition of
equivariance for statistical functionals. Comparisons in terms of breakdown
points seem only useful when restricted to equivariant functionals and even
here the connection between breakdown and equivariance is a tenuous one.
