Addendum to the discussion of “Breakdown and groups”
P. Laurie Davies, Ursula Gather
The Annals of Statistics · 2006 · Volume 34 · Issue 3
DOI: 10.1214/009053606000000399
Abstract
In his discussion of Davies and Gather [Ann. Statist. 33 (2005) 977–1035] 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.
Keywords
62G35, 62H11, Breakdown, directional data, Equivariance
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