Breakdown and groups

By P. L. Davies, Ursula Gather. 2005

89 citations recorded by the source corpus.

The concept of breakdown point was introduced by Hodges (1967) and Hampel (1968, 1971) and still plays an important though at times a controversial role in robust statistics. It has proved most successful in the context of location, scale and regression problems. In this paper we argue that this success is intimately connected to the fact that the translation and affine groups act on the sample space and give rise to a definition of equivariance for statistical functionals. For such functionals a nontrivial upper bound for the breakdown point can be shown. In the absence of such a group structure a breakdown point of one is attainable and this is perhaps the decisive reason why the concept of breakdown point in other situations has not proved as successful. Even if a natural group is present it is often not sufficiently large to allow a nontrivial upper bound for the breakdown point. One exception to this is the problem of the autocorrelation structure of time series where we derive a nontrivial upper breakdown point using the group of realizable linear filters. The paper is formulated in an abstract manner to emphasize the role of the group and the resulting equivariance structure.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d7f05e7e255c1cd158caf3ad37b11031ddcece58

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Citations in the corpus

High-Breakdown Robust Multivariate Methods (2008, background): “For a recent discussion of the importance of equivariance in breakdown considerations, see Davies and Gather (2005).”
Robust machine learning by median-of-means: Theory and practice (2017, background): “For instance, in the regression problem, one can consider equivariant estimators [22, 21] and [62, p.92], that is estimators T invariant by aﬃne transformation of the data: for all a ∈ R d and λ ∈ R , Among equivariant estimators, the common sense heuristic applies : “when more than half data are…”
Influence functions of the Spearman and Kendall correlation measures (2010, background): “The rejoinder of Davies and Gather (2005) discusses the difficulties of finding an appropriate definition of breakdown point for correlation measures.”
Principal Components Analysis Based on Multivariate MM Estimators With Fast and Robust Bootstrap (2006, methodology): “Because in PCA the parameter space for the eigenvec tors is bounded, defining a breakdown point in this context is not straightforward (see, e.g., Davies and Gather 2005), so we do not consider breakdown of PCA methods here.”