8 H is to ric a l R e m a rk s a n d A lt e rn a tiv e A p p ro a c h e s

110

8 H is to ric a l R e m a rk s a n d A lt e rn a tiv e A p p ro a c h e s

8.1

O v e rv ie w

Outliers

Rejectionofoutliers:DanielBernoulli(1770),astronomers

Trimmedmean:1821Rejectionrules:Pierce(1852),...:Testextremeobservations(largest&smallest)forbeinganoutlier,Rejectiftestissignificant,averagetheremainingones.

111

RobustnessofTests

Fisher’sexacttheoryforthenormaldistr.−→“monopoly”.E.S.Pearson(1931):Non-robustnessofF-testforvariances−→Focusonrobustnessofthelevel,sometimesneglectingpower

Non-parametrictests:PowerofWilcoxontestisverygood!

Robustestimation

Non-robustnessofthemean:Tukey(1960)Minimaxtheory:Huber(1964)InfluenceFunctionandBreakdownpoint:Hampel(1968)

112

8.2

O th e r A p p ro a c h e s a n d R e su lt s

aRejectionrulesseeabove

bL-estimatorsPrototype:TrimmedmeanWeightedaverage,weightbasedonrank:

Tn= X

i aiX[i] ,X[1] ,...,X[n] ordered, X

i ai=1Functional:Defineaibyafunctionh:[0,1]→|R

ai= Z

[(i−1)/n,i/n] hhzidz . Z

[0,1] hhzidz

ThFi= RxhhFhxiidFhxiRhhzidz

113 .2

cR-estimatorsPrototype:Hodges-Lehmannestimator,derivedfromSignedRankTestTeststatistic−→p-value.Chooseµsuchthatp-value=1.

−→bµ=medh≤ih(Xh+Xi)/2i,medianof“Walshaverages”

Generalform:Foragivenskewsymmetricfunctionhh.i,

ThFiissolutionofZh 12 Fhyi+ 12 (1−Fh2t−yi) dFhyi=0fort.Theintegralgets0whenthesampleX1,..,Xnanditsmirrorimageaboutt,2t−X1,..,2t−Xnhavetheirranks“mixed”aswellaspossible.

1148.2

dAdaptiveestimatorsForn→∞,thedistributioncanbeestimatedfromthedata

−→asymptoticallyfullyefficientestimators!Needlargesamples.

eRobustnesstowardsdependencelongrangedependence:Hampel(1987),Beran(1992),...

fOtherAspectsofRobustnessRobustifiedLikelihoodRatioTest,ConfidenceInt.:Huber(1968)Capacities:Huber&Strassen(1973)

115.2

gFurtherDevelopmentsPrincetonRobustnessYear1970:Simulationformanylocationandscaleestimators

Firstresultsonregression:Huber(1971)CovarianceMatrix:Maronna(1976)TimeSeries:Martin(1979),K

¨unsch(1981)

Robusttests(infinitesimal):Ronchetti(1980)

Combinatorialestimators:Rousseeuwandhisstudents

116 8.2

hMainResearchGroups

•Tukey,BellLabs&Princeton

−→St.Morgenthaler(Lausanne),DavidTyler(Rudgers)

•HuberandHampelinZurich−→AlfioMarazzi(Lausanne)Ronchetti,Rousseeuw,Stahel,K

¨unsch,M Function” M-estimation,Hampel’sApproach“basedontheInfluence ¨achler(ETH)

•ElvezioRonchetti,Geneva−→Maria-PiaVictoria-Feser,EvaCantoni,StephaneHeritier

•PeterRousseeuw,Belgium−→MiaHubert,C.Croux

117

•Argentina:VictorYohai,RicardoMaronna(&DougMartin,Seattle)Highbreakdownest.,MinimaxBias,soundtheory!−→Book

•PennState:Hettmansperger−→Shrader&McKean(Aus)R-estimators

•Oja,Finland(connectedtoBerne):Multivariatecombinatorial,nonparametric

•Chicago:Portnoy,Koenker,X.HeRegressionQuantiles,soundtheory

•...

118 8.2

iBooksHuber(1981(2009))Hampel,Ronchetti,Rousseeuw,Stahel(1986)Rousseeuw&Leroy(1987):RegressionMaronna,Martin,Yohai(2006)

Conferences

RobustnessYearinPrinceton(1970)Minneapolis(1989)

L1norm(Neuchatel,1987ff)Int.ConferenceonRobustStatistics(ICORS)