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1、LeastMedianofSquaresRegressionPETERJ.ROUSSEEUW*ClassicalleastsquaresregressionconsistsofminimizingcamefromEdgeworth(1887),improvingaproposalofthesumofthesquaredresiduals.Manyauthorshavepro-Boscovich.HisleastabsolutevaluesorL1criterionisducedmorerobustversio
2、nsofthisestimatorbyreplacingthesquarebysomethingelse,suchastheabsolutevalue.minimizeCIriI.Inthisarticleadifferentapproachisintroducedinwhichi=lthesumisreplacedbythemedianofthesquaredresid-Thisgeneralizesthemedianofaone-dimensionalsampleuals.Theresultingesti
3、matorcanresisttheeffectofnearlyand,therefore,hastobemadeunique(Harter1977).But50%ofcontaminationinthedata.Inthespecialcaseofwhereasthebreakdownpointofthesamplemedianissimpleregression,itcorrespondstofindingthenarrowest50%,itcanbeshownthatL1regressionyieldst
4、hesamestripcoveringhalfoftheobservations.GeneralizationsvalueE*=0asLS.AlthoughL1regressionprotectsarepossibletomultivariatelocation,orthogonalregres-againstoutlyingyi,itcannotcopewithgrosslyaberrantsion,andhypothesistestinginlinearmodels.valuesofxi=(xi],...
5、,xi,),whichhavealargeinfluenceKEYWORDS:Leastsquaresmethod;Outliers;Robust(calledleverage)onthefit.regression;Breakdownpoint.ThenextstepinthisdirectionwastheMestimator(Huber1973,p.800),basedontheideaofreplacingr?1.INTRODUCTIONin(1.l)byp(ri),wherepisasymmetri
6、cfunctionwithauniqueminimumatzero.Unlike(1.1)or(1.2),however,Theclassicallinearmodelisgivenbyyi=xil0++thisisnotinvariantwithrespecttoamagnificationofthexi,O,+ei(i=1,...,n),wheretheerroreiisusuallyerrorscale.Thereforeoneoftenestimatesthescalepa-assumedtobeno
7、rmallydistributedwithmeanzeroandrametersimultaneously:standarddeviationa.Theaimofmultipleregressionistoestimate0=(el,...,0,)'fromthedata(xi],...,xi,,yi).Themostpopularestimate6goesbacktoGaussorLegendre(seeStigler1981forarecenthistoricaldis-cussion)andcorres
8、pondstominimize2r?,i=lwhereIJJisthederivativeofpandxisasymmetricfunc-tion.(Findingthesimultaneoussolutionofthissystemofwheretheresidualsriequalyi-xilel-...-xipop.equationsisnottrivial,andinpracticeoneu
