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AsymptoticsforL1regressionestimatorsundergeneralconditionsKeithKnightDepartmentofStatisticsUniversityofTorontoAbstract:Itiswell-knownthatL1-estimatorsofregressionparametersareasymptoti-callyNormalifthedistributionfunctionhasapositivederivativeat0.Inthispaper,wederivetheasymptoticdistributionsundermoregeneralconditionsonthebehaviourofthedistributionfunctionnear0.SecondorderorweakBahadur-Kieferrepresentationsarealsoderived.1IntroductionConsiderthelinearregressionmodelYi= 0+ 1x1i+ + pxpi+i(1)where 0; 1; ; pareunknownparametersandfigareunobservableindependent,identicallydistributed(i.i.d.)randomvariableseachwithmedian0.Forsimplicity,wewillassumethatthexki’sarenon-randomalthoughtheresultswilltypicallyholdforrandomxki’s.WewillconsidertheasymptoticbehaviourofL1-estimatorsof =( 0; ; p);thatis,b 0;b 1; b pminimizetheobjectivefunctiongn( )=nXi=1jYi 0 1x1i pxpijoverall =( 0; ; p).TheasymptoticbehaviourofL1-estimatorsinregressioniswell-known,atleastinthecasewheretheerrorshaveadistributionfunctionF(t)whichisdi erentiableat0withthederivativepositive.Inparticular,ifwedenotethisderivativeby =F0(0),wehave(XTnXn)1=2(b )convergesindistributiontoa(p+1)-variateNormaldistributionwithmeanvector0andcovariancematrix(4 2) 1Iprovidedthatmax1 i nxTi(XTnXn) 1xi!0asn!1wherexTi=(1;x1i; ;xpi)andXnisthen (p+1)matrixwhosei-throwisxTi.(NotethatxTi(XTnXn) 1xi(i=1; ;n)aresimplythediagonalelementsoftheso-calledhatmatrix1Xn(XTnXn) 1XTn.)Ifn 1(XTnXn)!Cforsomepositivede nitematrixCthenitwillfollowthatpn(b )convergesindistributiontothe(p+1)-variateNormaldistributionwhosecovariancema-trixis(4 2) 1C 1.(See,forexample,BassettandKoenker(1978),Bloom eldandSteiger(1983),Baietal(1990)andPollard(1991)forvariousapproachestoprovingtheasymptoticNormality.)SecondorderresultsaregivenbyArcones(1996a,1996b),Babu(1989)andHeandShao(1996).Anaturalquestiontoaskiswhathappenswhenthedistributionfunctiondoesnothaveapositivederivativeat0.Whilethesecasesmayseempathological,theyare,infact,farfromit.Indeed,whileassumingtheexistenceofadensityseemsreasonable,itisanassumptionwhichisdi culttoverify.Infact,previousworksuggeststhattheasymptoticbehaviourofL1-estimatorsisverysensitivetothisassumption.Fori.i.d.observations,Smirnov(1952)identi esfourpossibletypesoflimitingdistributionsforsamplequantilesandcharacterizestheirdomainsofattraction.Jure ckov a(1983)considerstheasymptoticbehaviourofM-estimatorsoflocationundernon-regularconditions;herresultsincludetheL1estimatoroflocation(namelythesamplemedian)asaspecialcase.Onanotherfront,Arcones(1994)considerstheasymptoticbehaviourofso-calledLp-median(thatis,minimizersofPni=1jYi jp)for0p 1=2andshowsthattheconvergencerateisslowerthanOp(n 1=2).ToconsidertheasymptoticbehaviourofL1-estimators,wewillstartbyde ning(forsomesequenceofconstantsan), n(t)=Zt0pn(F(s=an) F(0))ds(2)whichforeachnisaconvexfunction.Ifthelimitoff n(t)gexistsforeacht,de ne (t)=limn!1 n(t):(3) (t)(ifitexists)isaconvexfunctiontakingvaluesin[0;1];notethat (t)mayequal1althoughtypically (t)willbe nite.(SeeExamples3and4insection3forcaseswhere (t)=1forcertaint.)Theexactformof in(3)canbemoreeasilyobtainedbyconsideringthelimitofpn(F(t=an) F(0));iflimn!1pn(F(t=an) F(0))=(t)(4)thentypically (t)=Zt0(s)ds:InthecasewhereF(x)isdi erentiableatx=0(withF0(0)0)thenan=pnand(t)= twhere =F0(0)andso (t)= t2=2.Moregenerally,condition(4)includescaseswhereFhasone-sidedderivativesat0((t)= +tfort0and(t)= tfort0wherean=pn)orisregularlyvaryinginaneighbourhoodof0.TheseconditionsareverysimilartothosegivenbySmirnov(1952).TheseassumptionsaresomewhatweakerthanthoseusedinJure ckov a(1983)2forthelocationcase.Inparticular,noticethatitisnotnecessarytoassumethatFisabsolutelycontinuous(withrespecttoLebesguemeasure);infact,Fcancontaindiscretecomponents.Wewillshowthat(undersuitableregularityconditionsonthedesign)an(b n )convergesindistribution.Todothis,wewill rstmodifytheobjectivefunctiongnasfollows:Zn(u)=anpnnXi=1hji xTiu=anj jiji:(5)ItiseasytoseethatthevectorbunwhichminimizesZnissimplyan(b n ).IfonenowregardsfZngasasequenceofrandomconvexfunctionsonRp+1andifthe nitedimensionaldistributionsofZn(u)convergeindistributiontothoseofsomefunctionZ(u)whichhasauniqueminimumUthenitwillfollowthatUn=an(b n )!dU=argmin(Z)asn!1(seeHj rtandPollard,1993;Geyer,1996).2LimitingdistributionsWewillnowformallystatetheregularityconditionsneededto ndthelimitingdistributionoftheL1-estimator.(A1)figarei.i.d.randomvariableswithmedian0withdistributionfunctionFcontinuousat0.(A2)Forsomepositivede nitematrixC,limn!11nXTnXn=C:(A3)Foreachu,limn!11nnXi=1 n(uTxi)= (u)forsomeconvexfunction (u)takingvaluesin[0;1]wheref n(t)gisde nedasin(2)(forsomesequencefang).Atthispoint,itisworthmakingafewcommentsontheregularityconditions.Condition(A2)isstandardandimplies,forexample,that1nmax1 i nxTixi!0:(A3)issimilarinspiritto(A2);itisessentiallyanothermomentconditionforthexi’s.If (t)(de nedin(3))is niteforalltthen (u)in(A3)cansometimesbeevaluatedas (u)=limn!11nnXi=1 (uTxi)3(assumingtheconvergenceof nto issu cientlyuniform).Ifthisisth