Estimators relies at the core on the rate of

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estimatorsrelies(atthecore) ontherateofconvergenceofthesampleminimumbeing)whilethatoftheestimatorsof"regular"parametersis0(n"J).Forconcreteness,weconsiderthehypothesesHo:(»/,</)=(»?o.<^0)HA:notHq.andthecorrespondinglikelihoodratioteststatisticwhich,byTheorem9,isasymptoticallyindependentofn{dd).nA(»/oi(io)=2log[A(»/o,do)]=2log<.(4.24)sup^{v,d)eeo"Wemay,asbefore,express(4.24)usingtheTaylorexpansiongivenin(3.16).
85TheregularportionoftheMLEbehavesidenticallyinthetruncationandcensoringproblems.Thatistosay\/n{Tf—Ti)isasymptotically normalwithmean0andcovariancematrixequaltotheinverseoftheFisherinformationThispropertymakesforthenecessarycancellationinexpression(.3.19)tocausethecorrespondingpartoftheTaylorexpansiontohaveanasymptoticxldistributionalongthelinesoutlinedbyJohnsonandWichem(1992,p.140).Regardlessiftheobservationsarecensoredortruncated,thefirstorderthresholdterminTaylorexpansion(3.16)isdistributedasa^xlExp(l))randomvariableinthelimit.Theorem11(DistributionofthelineartermforthecensoringparameterinaTaylorexpansionof£.(?/,</))AssumeXi,...,Xnareindependentandidenticallydistributedfromadistributionwithdensity(4-1)•IfVisconsistent,then(4.25)n0(4.26)convergesindistributiontoanexponentialrandomvariablewithmeanone.PROOFofTheorem11Wewritethevariable(4.26)asnn{d-d)^—logff,,d{xi)_=(rf-rf)53^1og{F^(</)l{^-_oo}+/ii(i)l{x.>j}}d(4.27)NowIMpUd)Ff,{d)Fr,{d)
86bytherightcontinuityof/,,(x)inx,thecontinuityof/,,(x)inri,thecontinuityofFr,{d)in(i/,d)andtheconsistencyofbothifandd.Further,n-cc{dd)=d)TIrand'^-oop,PnjjLJ)ItisaconsequenceofTheorems3and8thatn^{d-d)AYwhereVisexponentialwithmeanSon^{d-d)^^^AWwhere[Visexponentialwithmean\ w ) v - p j \ u )aAsintheanalysisofthetruncatedsituation,thefirstorder17-termsinaTaylorexpansionof(4.24)arezerobecauseifisthesolutiontothelikelihoodequation..A.lso,allthirdorderandhighertermsalongwithmi.xedsecondordertermsareasymptoticallyzerobecausetheirestimatorsconvergeatarate(whenyoumultiplythemtogether)fasterthan0(n~').Wehavealreadyaddressedthefirstordercensoringandsecondorder"regular"termsabove.Procedurallyspeaking,"testing"forthecensoredproblemisthesameas"testing"forthetruncatedproblem.Onesimplycomputestheexpressionin(4.24)andcomparestheresulttoapercentileofaxl+2'however,identicalinthesensethattakingthe"realized"data(thex,->—cxd)fromacensoreddistributionandtreating
87themasiftheywereasampleofsizefromatruncateddistribution(andignoringthecountn_oo)isnot,generally,thesameastakingintoaccountthecensoredobservationsbyusingrepresentation(4.2) in(4.24).Thisisobviousifyouconsiderthatinformationwouldbelostbysuchaprocedure.Asymptoticallytheinformationfromarandomsampleofsizendistributedaccordingto(4.1)isn(Vart(X)pEVar[i(-V)|.Y<</]).

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