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RobustStatistics
PartialOverview,GeneralPrinciples
WernerA.Stahel,ETHZurich
FallSemester2012
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1 In tr o d u c tio n
1.1
W h a t is R o b u st S ta tis tic s?
Example
Regression:Waterflowofariverat2locations.Simpleregression,13observations
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WaterFlow
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Libby
Newgate
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RegressionLine(LeastSquares)
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Libby
Newgate
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RobustlyEstimatedRegressionLine
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Libby
Newgate
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RegressionLineafterdroppingtheoutlier,LeastSquares
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Libby
Newgate
Least SquaresRobust MethodLSQ without outlier
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MathematicalStatistics:
•Modeldefinesdistributionofvariables,parameterstobeestimated/tested
•Theorydeterminesoptimalproceduresundertheconditionsdefinedbythemodel
•Inpractice,conditionsarenotsatisfied,butmay“holdapproximately”.
Robuststatisticsisthestatisticsofapproximateparametricmodels.
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1.2
H is to ry 1
Outliers
Rejectionofoutliers:DanielBernoulli(1770),astronomers
Trimmedmean:1821:Drop(k)largest&smallestobs.,averagetheremainingones.
Rejectionrules:Pierce(1852),...:Testextremeobservations(largest&smallest)forbeinganoutlier,Rejectiftestissignificant,averagetheremainingones.
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RobustnessofTests
Fisher’sexacttheoryforthenormaldistr.−→“monopoly”.E.S.Pearson(1931):Non-robustnessofF-testforvariancesNon-parametrictestsFocusonrobustnessofthelevel,sometimesneglectingpower(butWilcoxontestisgood!)
Robustestimation
Non-robustnessofthemean:Tukey(1960)Minimaxtheory:Huber(1964)InfluenceFunctionandBreakdownpoint:Hampel(1968)Otherapproaches,seelater
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Books
Huber(1981)−→Huber,Ronchetti(2009)Hampel,Ronchetti,Rousseeuw,Stahel(1986)Maronna,Martin,Yohai(2006)
Rousseeuw,Leroy(1987)Regression
