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1、LeastMedianofSquaresRegressionPETERJ.ROUSSEEUW*ClassicalleastsquaresregressionconsistsofminimizingcamefromEdgeworth(1887),improvingaproposalofthesumofthesquaredresiduals.Manyauthorshavepro-Boscovich.HisleastabsolutevaluesorL1criterionisducedmorerobustversionsofthisestimatorbyreplacingth
2、esquarebysomethingelse,suchastheabsolutevalue.minimizeCIriI.Inthisarticleadifferentapproachisintroducedinwhichi=lthesumisreplacedbythemedianofthesquaredresid-Thisgeneralizesthemedianofaone-dimensionalsampleuals.Theresultingestimatorcanresisttheeffectofnearlyand,therefore,hastobemadeuniq
3、ue(Harter1977).But50%ofcontaminationinthedata.Inthespecialcaseofwhereasthebreakdownpointofthesamplemedianissimpleregression,itcorrespondstofindingthenarrowest50%,itcanbeshownthatL1regressionyieldsthesamestripcoveringhalfoftheobservations.GeneralizationsvalueE*=0asLS.AlthoughL1regression
4、protectsarepossibletomultivariatelocation,orthogonalregres-againstoutlyingyi,itcannotcopewithgrosslyaberrantsion,andhypothesistestinginlinearmodels.valuesofxi=(xi],...,xi,),whichhavealargeinfluenceKEYWORDS:Leastsquaresmethod;Outliers;Robust(calledleverage)onthefit.regression;Breakdownpo
5、int.ThenextstepinthisdirectionwastheMestimator(Huber1973,p.800),basedontheideaofreplacingr?1.INTRODUCTIONin(1.l)byp(ri),wherepisasymmetricfunctionwithauniqueminimumatzero.Unlike(1.1)or(1.2),however,Theclassicallinearmodelisgivenbyyi=xil0++thisisnotinvariantwithrespecttoamagnificationoft
6、hexi,O,+ei(i=1,...,n),wheretheerroreiisusuallyerrorscale.Thereforeoneoftenestimatesthescalepa-assumedtobenormallydistributedwithmeanzeroandrametersimultaneously:standarddeviationa.Theaimofmultipleregressionistoestimate0=(el,...,0,)'fromthedata(xi],...,xi,,yi).Themostpopularestimate6goes
7、backtoGaussorLegendre(seeStigler1981forarecenthistoricaldis-cussion)andcorrespondstominimize2r?,i=lwhereIJJisthederivativeofpandxisasymmetricfunc-tion.(Findingthesimultaneoussolutionofthissystemofwheretheresidualsriequalyi-xilel-...-xipop.equationsisnottrivial,andinpracticeoneu