The notion of breakdown point was introduced by Hampel (1968, 1971) and has since played an important role in the theory and practice of robust statistics. In Davies and Gather (2004) it was argued that the success of the concept is connected to the existence of a group of transformations on the sample space and the linking of breakdown and equivariance. For example the highest breakdown point of any translation equivariant functional on the real line is 1/2 whereas without equivariance considerations the highest breakdown point is the trivial upper bound of 1. --

Davies, P. Laurie

Gather, Ursula

English

Research Papers in Economics

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Davies, P. Laurie; Gather, Ursula
Working Paper
Breakdown and Groups II
Technical Report / Universität Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten
Datenstrukturen, No. 2004,59
Provided in cooperation with:
Technische Universität Dortmund
Suggested citation: Davies, P. Laurie; Gather, Ursula (2004) : Breakdown and Groups II,
Technical Report / Universität Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten
Datenstrukturen, No. 2004,59, http://hdl.handle.net/10419/225721
Breakdown and Groups II 2
By P. L. Davies and U. Gather
University of Duisburg-Essen, University of Dortmund
This 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 rˆ ole in the theory and practice of
robust statistics. In Davies and Gather (2004) it was argued that the success of the
concept 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 highest
breakdown point of any translation equivariant functional on the real line is 1/2
whereas without equivariance considerations the highest breakdown point is the
trivial upper bound of 1. The situation considered in Davies and Gather (2004)
requires the existence of “banned” parameter values such as 1 in the case of trans-
lation and 0 and 1 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 parameter
values but that one may nevertheless wish to have some concept of breakdown. The
immediate example is that of directional data (see Mardia (1972)) where there is no
banned direction but a concept of breakdown might prove useful. It may seem that
1Received??
AMS 2000 subject classiﬁcations. Primary 62G07; secondary 65D10, 62G20.
Key words and phrases. equivariance, breakdown point, robust statistics
2Research supported in part by Sonderforschungsbereich 475, University of Dortmund.
12 ??
breakdown can be deﬁned here as the smallest amount of contamination required
to 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 breakdown
could be said to occur when the ﬁrst principal direction is orthogonal to the ﬁrst
principal 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 probability
distributions over S: We take P to be equipped with a metric d and deﬁne the
breakdown point by
(1.1) "¤(T;P;d) = inff" > 0 : jT(P) ¡ T(Q)j = ¼ for some Q with d(P;Q) < "g
where we have measured angles in radians. The problem with this deﬁnition is that
there may exist a sequence Qj;j = 1;:::N with say d(P;Qj) < j"=N and T(Qj) =
T(P)+j¼=N. In other words we can move from P to Q in small steps Qj and such
that at each stage the value of T(Qj) is perfectly reasonable for the distribution
Qj. An explicit example is given in the rejoinder of Davies and Gather (2004) in
the case of correlation. If breakdown is deﬁned in terms of banned parameter values
then such a construction is not possible. A deﬁnition of breakdown which includes
directional data has been given by He and Simpson (1992). They consider gross error
neighbourhoods and deﬁne breakdown to be the smallest neighbourhood within
which all parameter values are attainable. We feel however the deﬁnition of He and
Simpson as well as that of (1.1) refer to properties of a functional better described
in terms of lack of continuity rather than in terms of breakdown. Nevertheless there
is a situation which we think can be described by breakdown, namely when it is not
possible to deﬁne the functional in a consistent manner. The obvious example is
that of the mean Tm(P) =
R
xdP(x) for distributions P on R which is only deﬁned
for distributions P satisfying
R
jxjdP(x) < 1: This contrasts with the median?? 3
which can be deﬁned in a unique manner for all distributions on the real line and
still be aﬃne equivariant.
2. Breakdown and invariant distributions
2.1. 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 ! X
with identity ¶: We denote the set of all probability distribution P on X by P
which is equipped with a metric d: Let Θ be some parameter space equipped with
a group structure HG; induced by G; consisting of elements hg;g 2 G; for which
hg1 ± hg2 = hg1±g2. A functional T : PT ! Θ with Pt ½ P is called equivariant if
the following hold:
(a) PT is closed under all g 2 G;
(b) T is well deﬁned on PT;
(c) T(Pg) = hg(T(P)) for all P 2 PT and g 2 G:
This leads to the following deﬁnition of breakdown
(2.2) "¤(T;P;d) = inff" > 0 : d(P;Q) < " for some Q = 2 PTg
with of course "¤(T;P;d) = 0 if P = 2 PT: We note that this concept of breakdown
does not require a topology on the parameter space Θ: As an example we consider
the mean Tm which, as mentioned at the end of the last section, is deﬁned only
for distributions with a ﬁnite absolute ﬁrst moment. As any neighbourhood of any
distribution P on the real line contains distributions with an inﬁnite absolute ﬁrst
moment it follows that the mean has a breakdown point of zero. In contrast the
median can be well-deﬁned for all distributions on the real line and consequently
has a breakdown point of one in the sense of (2.2) above.4 ??
Suppose that there exist distributions P with Pg = P for some g with hg 6= h¶:
We denote the set of all such distributions by Pginv: If T is equivariant and g 2 Pginv
we have T(P) = T(Pg) = hg(T(P)) which is not possible as hg 6= h¶: We see that
(2.3) Pginv ½ P n PT
for every equivariant functional T: This implies
(2.4) "¤(T;P;d) · inff" > 0 : d(P;Q) < " for some Q 2 Pginvg:
We note that (2.4) gives an upper bound for the breakdown point which is the same
for all equivariant functionals T:
2.2. Finite sub-groups Suppose G contains a ﬁnite sub-group Gk of order k ¸ 2
so that gk = ¶ for all g 2 Gk: For any distribution P we set
(2.5) Pk =
1
k
k¡1 X
j=0
Pg
j
:
Then P
g
k = Pk so that Pk 2 Pginv and hence
"¤(T;P;d) · d(P;Pk):
If the metric d satisﬁes (2.1) and (2.2) of Davies and Gather (2004) we have
d(P;Pk) ·
k ¡ 1
k
d(P; ˜ Pk) ·
k ¡ 1
k
where
˜ Pk =
1
k ¡ 1
k¡1 X
j=1
Pg
j
:
Examples of sample spaces and groups G with subgroups of order k = 2 are the
unit circle and the unit sphere. In both cases the maximum breakdown point of any
direction functional is 1=2:?? 5
2.3. Total variation and ﬁnite sample breakdown points Although the above
results can be extended to ﬁnite sample breakdown points this may not always
make sense. In particular if P is an empirical measure then the breakdown point
measured using the total variation metric d = dtv may be reduced from 1=2 to 1=n
by the smallest of alterations in the values of the data. The same applies to the
ﬁnite sample breakdown points. This is the only example we know where the use
of a metric which allows for minor alterations in the values of the data points leads
to a completely diﬀerent breakdown point.
REFERENCES
Davies, P. L. and Gather, U. (2004). Breakdown and groups (with discussion). Annals of
Statistics 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.

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