Ursula Gather | Searcharxiv

arXiv2006

Addendum to the discussion of "Breakdown and groups"

In his discussion of Davies and Gather [Ann. Statist. 33 (2005) 977--1035] [math.ST/0508497] Tyler pointed out that the theory developed there could not be applied to the case of directional data. He related the breakdown of directional functionals to the problem of definability. In this addendum we provide a concept of breakdown defined in terms of definability and not in terms of bias. If a group of finite order $k$ acts on the sample space we show that the breakdown point can be bounded above by $(k-1)/k$. In the case of directional data there is a group of order $k=2$ giving an upper bound of 1/2.

math.ST↗

arXiv2005

Breakdown and groups

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.

math.ST↗