Kein Folientitel

1
Breakdown Points in Structured Models
Ursula GatherDepartment of Statistics
University of Dortmund (joint work with P.L.
Davies, Essen)
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Content

• ?. Definition of Breakdown Point first Examples
• ??. Group Structure and Main Theorem
• ???. Classical Example
• ?V. Examples Structured Models
• V. Remarks and Conclusion

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?. Definition of Breakdown Point
4
(
)

e
T

functional
a
of

D
d,
P,
T,


The
point

breakdown

Î
D

and

d
to

respect
with


P

at
P

by

defined
is
ü
ì
ï
ï
(
)
(
)
(
)
(
)



gt
e

e
Q
T
,
P
T
D
sup

0
inf
D
d,
P,
T,
ý
í
(
)
ï
ï
e
lt


Q
,
P
d
þ
î
Î


Q
,
P
P
local concept
5
ì
ï
ï
k
(
)
(
)
(
)
(
)



Q
T
,
P
T
D
sup


min
D
,
T,

.
p
.
b
.
s
.
f
x
í
k
n,
n
n
n
ï
ï


n
k
1
î
Q
k
n,
1
å
d




Q

where
y
k
n,
i
n
(
)
K


y
,
,
y

and
y
is a replacement sample containing at least n
- k points of xn
n
1
n
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Simple Examples
7
(
)


Q
,
0
8
Attempt at a general theory which explains these
examples
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??. Group Structure and Main Theorem
group of transformations on X
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??. Group Structure and Main Theorem
group of transformations on X
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Let denote ? - unit element of G G1
?(P)
sup P(B) B ? B (X ), gB ?B for some g? G1
Theorem If G1 ? ? for all G-equivariant T
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???. Classical Example
Scatter functionals on
Take
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Scatter functionals on
Take
Then
(Davies, 1993)
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Take
G ? , g g (x) m - x
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Take
G ? , g g (x) m - x
Then
G1 ?, i.e. and
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?V. Examples - Structured Models
Set
, ? gt 0
For each such
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Take
? space of finite distribution functions F on
(-?,?
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Take
? space of finite distribution functions F on
(-?,?
G1 ? ?
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But
, a, c, x ? (0, ?), (a, c) ? ?
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e.g.
G-equivariant
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Model
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HG ? ? ? transformation induced by G
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All conditions fulfilled except that
G1 ?
Just define T P ? ? by T(P)?0
T is equivariant w.r.t. G and
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• y-values of 1
• y-values of 0

with replaced by
is linearly independent, solutions of (?) will
remain bounded for all replacement samples
which contain at least k1 points of xn.
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sometimes 0 regarded as breakdown
However
describes non-relatedness of X and Y
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growth model
G-obvious
If y(t) gt 1 set y(t) 1, if y(t) lt 0 set
y(t) 0
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Definition of breakdown point should be
simple
reflect finite sample behaviour
allow meaningful comparisons between functionals
Example Genton Lucas (2003) for location in

?
?
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This definition is
not simple
does not reflect finite sample behaviour
- even if infinite, it collapses to 1 only
by inserting ?
does not permit meaningful comparisons between
functionals
and
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Definition of and
is
reflects finite sample behaviour
f.s.b.p.( ) and f.s.b.p.(MED)

T translation equivariant, f.s.b.p. ?
MED attains bound.

If no restrictions were imposed on T
breakdown point of 1 attainable, even with
perfectly sensible functionals
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reflects behaviour at
½
(1.5, 1.8, 1.3, 1.5?, 1.8?, 1.3?)
(1.5, 1.8, 1.3, 1.51?, 1.8?, 1.3?)
If ? ? ? MED( ), MED( ) break
down.
Behaviour of MED( ) not explained by
translation equivariance
and breakdown point of ½ . Invisible small
print?
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Conclusion
(X,B,P ) ? d D
G HG
Comparison w.r.t. breakdown
calls for G-equivariant T
Connection between breakdown and equivariance is
a tenous one.
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Sketch of proof B invariant under g
?
?
?
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III. Classical Example
Location summary in
Then
G1 G \ ? ?(P) 0 for all P ? P as
there is no set B ? B satisfying
gB g? , for some g ? G1
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????
If gB ?B , g ? G1 ?
?(P) mass of largest atom of P"
(already Davies, 1993)
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Take
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Then
(Davies, 1993)
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Attaining the bound?
Answer not that simple
Location functional in
for all translation equivariant location
functionals T
Attained by
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Replace
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• If in location case T not only translation
  equivariant but
• affine equivariant, is the bound of 1/2 still
  attainable?

Answer is still open, but one can construct T
which has highest possible finite replacement
breakdown point at some given empirical measure
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Attaining the bound continued
Define
with
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Using
but with different metric d3
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Metric d3
with Ij finite interval
(generalized Kuiper metric)
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• Semimetrics
• A canonical definition of D (local)

This gives a semimetric DP symmetric, fulfills
?-inequality, ... and all conditions are
satisfied, whenever d is a G-invariant
semimetric on P
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IV. Examples - Structured Models
Basic model stationary AR(1) process
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Stromberg and Ruppert (1992)
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Then