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Founded 1348Charles University
Johann Kepler University of LinzJohann Kepler University of Linz
FSV UK
STAKAN III STAKAN III
Institute of Economic Studies Faculty of Social Sciences
Charles UniversityPrague
Institute of Economic Studies Faculty of Social Sciences
Charles UniversityPrague
Jan Ámos VíšekJan Ámos Víšek
- BASIC IDEAS
ROBUST STATISTICS - ROBUST STATISTICS -
Austria, Linz 16. – 18. 6.. 2003
- BASIC IDEAS
Schedule of today talk
A motivation for robust studies
Huber’s versus Hampel’s approach
Prohorov distance - qualitative robustness
Influence function - quantitative robustness • gross-error sensitivity
• local shift sensitivity • rejection point Breakdown point
Recalling linear regression model
Scale and regression equivariance
Introducing robust estimators
continued
Schedule of today talk
Maximum likelihood(-like) estimators - M-estimators
Other types of estimators - L-estimators -R-estimators - minimum distance - minimum volume
Advanced requirement on the point estimators
AN EXAMPLE FROM READING THE MATH
.8
1lim
8
xx
.5
1lim
5
xx
Having explained what is the limit,
an example was presented:
To be sure that the students
they were asked to solve the exercise :
The answer was as follows:
really understand what is in question,
Why the robust methods should be also used?
Fisher, R. A. (1922): On the mathematical foundations of theoretical statistics.
Philos. Trans. Roy. Soc. London Ser. A 222, pp. 309--368.
)1(
61
)x(var
)x(var
n)(t
n)1,0(N
nlim
)1(
121
)s(var
)s(var2n)(t
2n)1,0(N
nlim
Continued
Why the robust methods should be also used?
9t 5t 3t
nx
2ns
0.93 0.80 0.50
0.83 0.40
)T(var
)T(var
n)(t
n)1,0(N
nlim
0 !
)s(var 2nt 3
)s(var 2n)1,0(N
is asymptotically
infinitely larger than
Standard normal density
Student density with 5 degree of freedom
Is it easy to distinguish between normal and student density?
Continued
Why the robust methods should be also used?
New York: J.Wiley & Sons
Huber, P.J.(1981): Robust Statistics.
n
1i in xxn2
d 2
1
n
1i
2in xx
n
1s
)(
3)x()1()x(F
n2
n
n2
n
n dE/dvar
sE/svarlim)(ARE
0 .001 .002 .05
.876 .948 1.016 2.035
)(ARE
Continued
Why the robust methods should be also used?
So, only 5% of contamination makes two times better than . ns
nd
Is 5% of contamination much or few?
E.g. Switzerland has 6% of errors in mortality tables, see Hampel et al..
Hampel, F.R., E.M. Ronchetti, P. J. Rousseeuw, W. A. Stahel (1986):
Robust Statistics - The Approach Based on Influence Functions. New York: J.Wiley & Sons.
Conclusion: We have developed efficient monoposts which however work only on special F1 circuits.
A proposal: Let us use both. If both work, bless the God. We are on F1 circuit. If not, let us try to learn why.
What about to utilize, if necessary, a comfortable sedan.
It can “survive” even the usual roads.
Huber’s approach
One of possible frameworks of statistical problems is to consider
a parameterized family of distribution functions.
Let us consider the same structure of parameter space but instead of each distribution function
let us consider a whole neighborhood of d.f. .
Huber’s proposal:
Finally, let us employ usual statistical technique for solving the problem in question.
continued - an exampleHuber’s approach
Let us look for an (unbiased, consistent, etc.) esti- mator of location with minimal (asymptotic)
variance for family . )x(F)x(F
, i.e. consider instead of single d.f. the family .
H )x(H:)x(H)x(F)1()x(GQ H,
F Q
Let us look for an (unbiased, consistent, etc.) estimator of location with minimal (asymptotic) variance
for family of families .
Q)x(G H,
Finally, solve the same problem as at the beginning of the task.
For each let us define
Hampel’s approach
The information in data )x,,x,x( n21 x
is the same as information in empirical d.f. .nF
An estimate of a parameter of d.f. can be then considered as a functional .)F(T nn
has frequently a (theoretical) counterpart .)F(TAn example:
)F(TdFxxn
1x nn
n
1i i
)F(T)x(dFxXE
)F(T nn
continued Hampel’s approach
Expanding the functional at in direction to , we obtain:
)F(T n FnF
nnnn R)x(dF)x(dF)x,F('T)F(T)F(T
where is e.g. Fréchet derivative - details below.)x,F('T
Message: Hampel’s approach is an infinitesimal one, employing “differential calculus” for functionals.
Local properties of can be studied through the properties of .)F('T
)F(T nn
Qualitative robustness
Let us consider a sequence of “green” d.f. which coincide with the red one,
up to the distance from the Y-axis . n/1
Does the “green” sequence converge to the red d.f. ?
Let us consider Kolmogorov-Smirnov distance, i.e.
continuedQualitative robustness
)x(F)x(Fmax)F,F(d nRx
n
K-S distanceof any “green” d.f.
from the red one is equal to the length of yellow
segment.
The “green” sequence does not converge in K-S metric
to the red d.f. !
CONCLUSION:Independently on n,
unfortunately.
continuedQualitative robustness
A , ) A( G ) A( F; inf ) G, F(
Prokhorov distance
Now, the sequenceof the green d.f. converges
to the red one.
We look for a minimal length, we have to move the green d.f.
- to the left and up - to be above the red one.
In words:
CONCLUSION:
)( )(),( nGnF TT)G,F( LL
Conclusion : For practical purposes we need something “stronger” than qualitative robustness.
:G,F00 DEFINITION
E.g., the arithmetic mean is qualitatively robust at normal d.f. !?!
In words: Qualitative robustness is the continuity with respect to Prohorov distance.
i.i.d.
Qualitative robustness
)F(Tˆx,...,x,x nnn21
)( nF1 TF)x( LL
Quantitative robustness
nnnn R)x(dF)F,T,x(IF)F(T)F(T
ni
n
1i
2/1nn R)F,T,x(IFn))F(T)F(T(n
�
The influence function is defined where the limit exists.
Influence function
)F,T,x(IF lim0h h
)F(T)hF)h1((T x
continuedQuantitative robustness
Characteristics derived from influence function
)F,T,x(IFsup
Rx
*
Gross-error sensitivity
)F,T,y(IF)F,T,x(IFsup{
yx
*
Local shift sensitivity
/ }yx
rxfor0)F,T,x(IF;0rinf* Rejection point
Breakdown point
(The definition is here only to show that the description of breakdown which is below, has good mathematical basis. )
)F,ˆ( )n(*
1))K(βG(εG)π(F, (n) :compaktis)(K,R)(K:sup p
10
nfor
Definition – please, don’t read it
in the sense that the estimate tends (in absolute value ) to infinity or to zero.
is the smallest (asymptotic) ratio )F,ˆ( )n(*
which can destroy the estimate
In words
obsession
(especially in regression
– discussio
n below)
An introduction - motivation Robust estimators of parameters
Let us have a family )}x(f{
and data .n21 x,,x,x
Of course, we want to estimate .
Maximum likelihood estimators :
)x(fmaxargˆi
n
1i
)x(flogmaxarg i
n
1i
What can cause a problem?
What can cause a problem? Robust estimators of parameters
})x(2/1exp{)2()x(f 22/1
2)x()2log()x(flog2
}{2n
1i iR
)x(maxarg
}{ 0)x(argn
1i iR
nxn
1i i n
1i ixn/1
Consider normal family with unit variance: An example
2n
1i iR
)x(minarg
(notice that does not depend on ).So we solve the extremal problem
)2log(
A proposal of a new estimator
Robust estimators of parameters
Maximum likelihood-like estimators :
Once again: What caused the problem in the previous example?
So what about
kxfor)x(k
1)x( 2
kxforx
n
1i ixn/1
)x(maxarg i
n
1i
}{ 0)x(arg i
n
1i
2)x()x(
kxfor)x()x( 2
k
1
kxforx
Robust estimators of parameters
0
kxfor)x)(k/1()x()2/1(
kxfor1 x)x()2/1(
quadratic part
linear part
The most popular estimators
Robust estimators of parameters
maximum likelihood-like estimators
M )x(maxarg i
n
1i
M-estimators
based on order statistics
L )xw(maxarg )i(i
n
1i
L-estimators
based on rank statistics
R )Rw(maxarg ii
n
1i
R-estimators
Robust estimators of parameters The less popular estimators
but still well known.
Robust estimators of parameters
based on minimazing distance between empirical d.f. and theoretical one.
d )F,F(dminarg n
n
1i
Minimal distance estimators
based on minimazing volume containing given part of data and applying “classical”
(robust) method.
V }{ }Vx{Iw:)x(wminarg ii
n
1i ii
Minimal volume estimators
Robust estimators of parameters
The classical estimator, e.g. ML-estimator, has typically a formula to be employed for evaluating it.
Algorithms for evaluating robust estimators
Extremal problems (by which robust estimators are defined) have not
(typically) a solution in the form of closed formula.
To find an algorithm how to evaluate an approximation to the precise solution.
Firstly
To find a trick how to verify that the appro- ximation is tight to the precise solution.
Secondly
High breakdown point
obsession (especially in regression
– discussion below)
Hereafter let us have in mind that we speak implicitly about
Recalling the model
Put
1p,,2,1j,n,,2,1i,x)X( ijij
( if intercept n,,2,1i1x 1i ),T
n21 ),,,( .andTp21 ),,,(
where Tip2i1ii )x,,x,x(X .
0T
i
n
1j
0jiji XXY
Tn21 )Y,,Y,Y(Y
LineLineaar regresr regressionsion model model
0XY
So we look for a model“reasonably” explaining data.
LineLineaar regresr regressionsion model model
Recalling the model graphically
This is a leverage point and this is an outlier.
LineLineaar regresr regressionsion model model
Recalling the model graphically
Formally it means:
If for data )X,Y( the estimate is , )X,aY(than for data the estimate is .ˆa
Equivariance in scale
If for data )X,Y( the estimate is , )X,XY( Tthan for data the estimate is .ˆ
Equivariance in regression Scale equivariant
Affine equivariant
We arrive probably easy to an agreementthat the estimates of parameters of model
should not depend on the system of coordinates.
Equivariance of regression estimators
Unbiasedness Consistency
Asymptotic normality Gross-error sensitivity
Reasonably high efficiency Low local shift sensitivity
Finite rejection point Controllable breakdown point
Scale- and regression-equivariance Algorithm with acceptable complexity
and reliability of evaluation Heuristics, the estimator is based on,
is to really work
Advanced (modern?) requirement on the point estimator
Still not
exhaustive
THANKS for A
TTENTION