Addendum to the discussion of "Breakdown and groups"

Details Addendum to the discussion of "Breakdown and groups" Addendum to the discussion of "Breakdown and groups"

2005-07-05

arxiv.org/abs/math/0507089v2

Annals of Statistics 2006, Vol. 34, No. 3, 1577-1579

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000399 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000399

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.

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