Davies, P. L.

Gather, U.

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1Breakdown and Groups II 2By P. L. Davies and U. GatherUniversity of Duisburg-Essen, University of DortmundThis extends the work of Davies and Gather (2004).1. Introduction The notion of breakdown point was introduced by Hampel(1968, 1971) and has since played an important roˆle in the theory and practice ofrobust statistics. In Davies and Gather (2004) it was argued that the success of theconcept is connected to the existence of a group of transformations on the sam-ple space and the linking of breakdown and equivariance. For example the highestbreakdown point of any translation equivariant functional on the real line is 1/2whereas without equivariance considerations the highest breakdown point is thetrivial upper bound of 1. The situation considered in Davies and Gather (2004)requires the existence of “banned” parameter values such as∞ in the case of trans-lation and 0 and∞ in the case of scale. In the discussion of Davies and Gather (2004)Tyler pointed out that there are situations where there are no banned parametervalues but that one may nevertheless wish to have some concept of breakdown. Theimmediate example is that of directional data (see Mardia (1972)) where there is nobanned direction but a concept of breakdown might prove useful. It may seem that1Received??AMS 2000 subject classifications. Primary 62G07; secondary 65D10, 62G20.Key words and phrases. equivariance, breakdown point, robust statistics2Research supported in part by Sonderforschungsbereich 475, University of Dortmund.12 ??breakdown can be defined here as the smallest amount of contamination requiredto cause the direction to differ from the original direction by 180◦. A further exam-ple explicitly mentioned by Tyler is that of principal components where breakdowncould be said to occur when the first principal direction is orthogonal to the firstprincipal direction for the non-contaminated sample. To be more mathematical,consider the unit circle S and a direction functional T on the set P of probabilitydistributions over S. We take P to be equipped with a metric d and define thebreakdown point by(1.1) ε∗(T, P, d) = inf{ε > 0 : |T (P )− T (Q)| = pi for some Q with d(P,Q) < ε}where we have measured angles in radians. The problem with this definition is thatthere may exist a sequence Qj , j = 1, . . . N with say d(P,Qj) < jε/N and T (Qj) =T (P )+ jpi/N . In other words we can move from P to Q in small steps Qj and suchthat at each stage the value of T (Qj) is perfectly reasonable for the distributionQj . An explicit example is given in the rejoinder of Davies and Gather (2004) inthe case of correlation. If breakdown is defined in terms of banned parameter valuesthen such a construction is not possible. A definition of breakdown which includesdirectional data has been given by He and Simpson (1992). They consider gross errorneighbourhoods and define breakdown to be the smallest neighbourhood withinwhich all parameter values are attainable. We feel however the definition of He andSimpson as well as that of (1.1) refer to properties of a functional better describedin terms of lack of continuity rather than in terms of breakdown. Nevertheless thereis a situation which we think can be described by breakdown, namely when it is notpossible to define the functional in a consistent manner. The obvious example isthat of the mean Tm(P ) =∫x dP (x) for distributions P on R which is only definedfor distributions P satisfying∫ |x| dP (x) < ∞. This contrasts with the median?? 3which can be defined in a unique manner for all distributions on the real line andstill be affine equivariant.2. Breakdown and invariant distributions2.1. Definition of breakdown We use the notation of Davies and Gather (2004).Let X be a sample space and G a group of measurable transformations g : X → Xwith identity ι. We denote the set of all probability distribution P on X by Pwhich is equipped with a metric d. Let Θ be some parameter space equipped witha group structure HG , induced by G, consisting of elements hg, g ∈ G, for whichhg1 ◦ hg2 = hg1◦g2 . A functional T : PT → Θ with Pt ⊂ P is called equivariant ifthe following hold:(a) PT is closed under all g ∈ G,(b) T is well defined on PT ,(c) T (P g) = hg(T (P )) for all P ∈ PT and g ∈ G.This leads to the following definition of breakdown(2.2) ε∗(T, P, d) = inf{ε > 0 : d(P,Q) < ε for some Q /∈ PT }with of course ε∗(T, P, d) = 0 if P /∈ PT . We note that this concept of breakdowndoes not require a topology on the parameter space Θ. As an example we considerthe mean Tm which, as mentioned at the end of the last section, is defined onlyfor distributions with a finite absolute first moment. As any neighbourhood of anydistribution P on the real line contains distributions with an infinite absolute firstmoment it follows that the mean has a breakdown point of zero. In contrast themedian can be well-defined for all distributions on the real line and consequentlyhas a breakdown point of one in the sense of (2.2) above.4 ??Suppose that there exist distributions P with P g = P for some g with hg 6= hι.We denote the set of all such distributions by Pginv. If T is equivariant and g ∈ Pginvwe have T (P ) = T (P g) = hg(T (P )) which is not possible as hg 6= hι. We see that(2.3) Pginv ⊂ P \ PTfor every equivariant functional T. This implies(2.4) ε∗(T, P, d) ≤ inf{ε > 0 : d(P,Q) < ε for some Q ∈ Pginv}.We note that (2.4) gives an upper bound for the breakdown point which is the samefor all equivariant functionals T.2.2. Finite sub-groups Suppose G contains a finite sub-group Gk of order k ≥ 2so that gk = ι for all g ∈ Gk. For any distribution P we set(2.5) Pk =1kk−1∑j=0P gj.Then P gk = Pk so that Pk ∈ Pginv and henceε∗(T, P, d) ≤ d(P, Pk).If the metric d satisfies (2.1) and (2.2) of Davies and Gather (2004) we haved(P, Pk) ≤ k − 1kd(P, P˜k) ≤ k − 1kwhereP˜k =1k − 1k−1∑j=1P gj.Examples of sample spaces and groups G with subgroups of order k = 2 are theunit circle and the unit sphere. In both cases the maximum breakdown point of anydirection functional is 1/2.?? 52.3. Total variation and finite sample breakdown points Although the aboveresults can be extended to finite sample breakdown points this may not alwaysmake sense. In particular if P is an empirical measure then the breakdown pointmeasured using the total variation metric d = dtv may be reduced from 1/2 to 1/nby the smallest of alterations in the values of the data. The same applies to thefinite sample breakdown points. This is the only example we know where the useof a metric which allows for minor alterations in the values of the data points leadsto a completely different breakdown point.REFERENCESDavies, P. L. and Gather, U. (2004). Breakdown and groups (with discussion). Annals ofStatistics to appear.Hampel, F. R. (1968). Contributions to the theory of robust estimation. Ph. D. thesis, Dept.Statistics, Univ. California, Berkeley.Hampel, F. R. (1975). Beyond location parameters: robust concepts and methods (with discus-sion). Proceedings of the 40th Session of the ISI, Vol. 46, Book 1, 375–391.He, X. and Simpson, D. G. (1992). Robust direction estimation. Ann. Statist. 20, 351–369.Mardia, K. V. (1972). Statistics of directional data. Academic Press, London, New York.

Breakdown and Groups II

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Breakdown and Groups II

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