Testing for a Global Maximum of the Likelihood

oTCedex,estinginfomrtaotsGlobaldMaximumtheoryof1975),the(2000)LikaelihoMathematics,oapplicationsdknoChristopheoBiernaconsistenckimWhenindicationsevTheeralamongroinstanceotsthetokithe16likINTRODUCTIONelihoumovdmaequationtoexist,standardthethererotootCram?rcorrespTondingestorothesevglobalofmax-approacimizer(seeofhap.thetlikmethoelihoformoailable.dprofessor,isFgenerallyyretainedbiernac@math.univ-fcombutmanthistheproelihocedureissupped,osesthatthatballroplikossibleequation.royotsusareaidenrotied.likSince,equationinandmangeneralizationyandcases,generallytheoglobalwhicmaximizerisiscasetheroonlypapconsistenettvroforot,rowae1983,propincludingosefromaemplotestotoordetectwhenifforaagivChristopheenAssistansolutiontisersit?consistenhe-Comt.deThis25030testrancerelies1onInsomeynecessarywhereandmaximsucienliktoconditionsprincipleforinconsistencyolvofstatisticiansawrothereotyandesimplyultipleconsistsotsofthecomparingelihotheddierenceUnderbregularitetconditions,wtellseenthattiswuniqueotexpotectedthelog-likelihoelihodo(seed1946expressions.itsMonultidimensionalte-CarloinstudiesaroneandGruenhageabutrealgivlifepexam-orpleonshohwotthatconsistentheinpropofosederalproots.cedurereviewleadsertoSmallencouragingal.results.discussesInariousparticular,hesitselectingclearlytheoutpotserformsalsoanotherdiscussionaLehmannvcailable6),testforofiteratingthisconsistenkind,estimators,espyingeciallybforotstraprelativdelyexaminingsmallasymptoticssampleexplicitsizes.ulasKEYroWareORDS:vConsistency;AnotherMaximBiernacumisliktelihoDepartmenoofd;UnivLodecalrancandt?,globalroutemaximizers;GraT,estBesan?onpFo(E-mail:wte.fr).er.1.p
estossibilitty.isastothesimplywhenselectrothecurrenroaluatedotinleadingthistoultidimensional.theAnothermaximmethoumossiblelikeelihotheoindmatrixvvalueVsinceofWisald(1998)(1949)toestablishedaconsistencyequation.ofgivtheoglobalotmaximizer(1997)ofgotherolikeselihotheostatisticdductunderform.somehighlighconditionsextreme-v(ttedypicallyantheecomesglobalbmaximizerofisparameteraMarkrorandomotonalthoughreasonableityisofnotlikalwords,atoysshouldtrue,ofinus,particularforforitsomeisGaussianMortonmixturestoastonoticedtorsthbdyitsKieferotandandWw)olfohwitzw1956).FisherNoteparameteralsote-CarlothatofWeryald'sbasedpropasymptoticertiespofbthe(1991)maximconumhlikbelihonoofdsuppestimatorparameter(MLE)and/orareisgeneralizedconbetyoseWhitep(1982)dinotheautomaticallymorehrealistichcaseinwhereforthegivprobabilittoyomootherdelhisallomisspifecied.roSo,eexceptglobalinlikthefunction.rareiscaseslowherenewthetoMLEtilmaroyrejected.bHeydeehainconsistenosedty(seedness-of-texamplestheinotNeymankandforScottHessian1948elihoorehamorelikrecenectationtlytheinhand.Fspirit,erguson(1999)1982borhosealsodecisioninbasedStefanskibandtheCarrollof1987ectedamongoutothers),itsthethestrategyerimenwhicrestrictedhunidimensionalconsistsaofdselectingaluetheonglobalaluemaximizertheoryseemsAstooinbouteyaeallstraighintforweconometricardtext,proapproaccedurebtoimpracticalretainecauseantheadequateumroerot.computationsHothewortevtheer,spacesomelargepracticalthedicultiesspaceomccurInandtrast,watoueal.aimproptoaaddressstartingthemoininmethothebasedpresenbtotstrappapconstructer.aIndeed,searcinregion.practice,approacamasearcconsisthconstructingfortestallconsistencyroaotsencorrespotondingthetoeliholodcalInmaximizerswmasucyatakdewsconsid-decideerableatimeenandotnobguaranadoptedteeaismaximizergivtheenelihothatdallThloitcalpmaximizerstowillokhaavroeandbtesteenunfoundtheintaotnitenottime,Heydeevandenandif(1998)thevnpropumeitherbemploeraoforocriterionotsselectisbbrooundedor(seepicBarnettthe1966otforwhicantheexampleofoflog-likanounbbvoundedasymptoticallyneumexpbeveratofroroatots).InBeysameondGanthisJiangbasic(GJ99strategyshortofelosearcching,afewofpreviouswhicstudiesisareonadierencevetailable.eenFproorforminstance,theDeexpHaaninformation(1981)abproptheosedandaHessianpUnfortunately,-condenceMoninexptervtsaltheofcasetheamaximparameterumtlikveliho2o‘(?) ? ?0
? ?0
E r‘(?)=0:?0
ˆ?n
£ ⁄
r‘(?) =0:ˆ?=?n
ˆ? ‘(?)n
£ ⁄ £ ⁄
‘(?) ¡ E ‘(?) …0;ˆ ? ˆ?=? ?=?n n
£ ⁄
ˆE ‘(?) ?? n?=?0
ybulationethighlywnaturaleenularlytfollowconcludeodierenexpvectederimenlog-likparameterselihoSectionotodsomethingexpressions.handDenotingtermboyImplementoparameterorderoind.testfarthetolog-likGJ99'selihosetonewdloofnot.atermsparameterareGJ99'sestimatorswhosehtrueretainsvaalueEquations(unknomethown)yistforwtoearssimilarthe,oneittomabystudyexistandsomearevshortaluesacedureand,videddierenptcomparisonsfromotpromaximizeraw,itwhicbhthesatisfyofalsowtconsistenpresentheenowstatistic,er,basedpapathisroInwhicestimation.bparameterandultidimensionalthe(1)isThandus,mtheisproblemThroughisitthatthethereermaosedyerformsbGJ99'seammultiplemaronootsmeaningless.toorganizedtheData,liktoelihotheoteddwhereequationtation(globalismaximizer,ailable.loerimencalrealmaximizer,thenstationarySectionpaluateoinoftInandwsoroon),(aitcalmeansormelse)ultipleouldroIndeed,otsseemsmthatofothsituationinsuclefthsidethat(3)usualttheointeciallytespoftest,samethiswsimplyisiswhict.edtheondierenceemploinifelythetheofconsistenuseSo,thetuitivrecommend,totestdicultasisgoitd"(2)otInparameterorderhtoeriesdetectoththe(2)ro(3).ottingconsequence,newadAspartic-correspeasyondingapplicabilittotoaultidimensionalglobalcasesmaximizerstraighofard.sizes.expsamplets,smallappelythat,pthewideaofofpropthistestpapoutperthisisofvmethoeryAssimple:consequence,AconsiderglobalultidimensionalmaximizersituationswyouldesatisfywnotfromonlyThe(2),isbutasalsow.relativassumptionswiththeoreticalerolswbuiltotestppresenininlies2primarilyadierencepresenTheofelihotestwalsolikvt.Simtheexptotsotarodataloareconsistenproisinequation3devothewitherformanceofthetesttest.givparticular,InlastresultsaGJ99'sifaredecideen.thistheersection,ae3tpap(3)withwhereasdiscussion.aninconsistenX ;X ;:::;X n1 2 n
X f (x) f(x;?)t
? ? ?0
R
E lnf(X;?)= lnf(x;?)f (x)dx ?t t 0
Θ ?0
f(x;?)=0 ?2Θ
R
f(x;?) ? f(x;?)d·
’ (x;?) = rlnf(x;?) ’ (x;?) = lnf(x;?)¡ E lnf(X;?)1 2 ?
Pn
` (?) = ’ (X ;?)=n d (?) = E ’ (X;?) = E ` (?) j = 1;2 ’(x;?) = (’ (x;?);’ (x;?))j j i j t j t j 1 2i=1
Pn
`(?) = (` (?);` (?)) d(?) = (d (?);d (?)) ‘(?) = lnf(X ;?)1 2 1 2 ii=1
ˆ? X ;:::;X ? r‘(?)1 n n
` (?) v(?) = Var lnf(X;?) j¢j L1 ? 1
max E sup j’ (X;?)j<1 max B <1 B =E sup jr’ (X;?)jj=1;2 t j j=1;2 j j t j?2Θ ?2Θ
0<Var lnf(X;? )<1t 0
jrv(? )j<10
ˆ? ?n 0
ˆ` (? )2 n
pandtiableparameteretoesect2.1.haLethrespardswithandtiableLetdierensamewice(ftnoiswvvalueesisandanCONSTRUCTIONinoterior((c)with.sequel,pInoinalsobWeotheinlog-likTheelihotoodThefunctionofofthet.isbased2.onTHEobservwations),alleSettforwa.e..(b)as.theadensitm,ultidimensional.andmaximized.letparametric6presenparameter.tDeneonbtesteessenayrooneotandoffortheofvtoaluefamilyofonewhereasymptotic,inthewingsofolloequivthatalenuniquetlythataandroassumeotOFofTESTtheTtegralwithexptheoectedremslog-likbbindeptheoremsendentherandomundervhectorsothesisthecon(e)ergence.distributionofsahvwhicariableofvingspaceycompactfunctiona.is)isConsidertanandidenwiceisdieren-.tiableeandtdenotingwbwytheoremsunderwhictheourinreliestheantegraltialsign.anorm,.wrstegivassumenecessarythesucienfolloconditionswingconconditions:ergence(d)dFirst,densitletwusydene.spacesecondparametergivThethe(a)distributionhold:where,isconditionselihoyregularitgivthisinypendixof4varianceProalsoofossiblyoth.areFinallyen,AppdeningA.thevd(?)=(0;d (? )))?=? :2 0 0
Pˆ ˆ? ¡!? P(? 2Θ)!1n 0 n
Pˆ` (? )¡!d (? ):2 n 2 0
Pˆ? ¡!?n 0
? ¶
Var lnf(X;? )D t 0ˆ` (? )¡!N d (? ); :2 n 2 0
n
ˆ? ?n 0
f (x) f(x;?) f (x)=f(x;? )t t 0
ˆd (? ) = 0 ?2 0 n
Pˆ` (? )¡!02 n
ˆ` (? )2 n
v(? ) = Var lnf(X;? )0 ? 00
ˆ ˆ ˆv(? )=Var lnf(X;? ) ? ?n ˆ n 0 n?n
Pn 2ˆ ˆ ˆV (? ) = (lnf(X ;? )¡‘(? )=n) =nn n i n ni=1
ˆv(? )n
ˆv(? )n
ˆV (? )n n
ˆ ˆv(? ) V (? )n n n
ˆv(? ) v(? )n 0
ˆp ` (? )2 n D
nq ¡!N(0;1):
ˆv(? )n
ˆ ˆ ˆE lnf(X;? ) ` (? ) v(? )ˆ n 2 n n?n
ˆY Y f(x;? )1 m n
P Pm m 2ˆ ˆA = lnf(Y ;? )=m B = (lnf(Y ;? )¡A ) =mm i n m i n mi=1 i=1
tivt,and,alenothThenothesis.(4)evthattest.oseinequivformssuppelongsaddition,consequenInnisaresatised.proareto(a)-(d)estimatorconditionsythatinosegeneratedothesisectivSuppSupp1.fromemtestorothesis,Theonewhereoexptoectationsthat,areconestimatedWinarianceant,empiricalewmethoavya.,ApptheendixothBydiscussesdistributionproposeerties.ofnotheseintunderwhoeestimators:tobaothwingaretconsistenerformtisbut(7)somepractice,empiricalforobservergenceationsardsshoconsiderwthethatthetheandrstpone,yypestimatedhMonullifnnotasThand,w,samplehas.ahloullwwhereerandmeanaresquaredestimatederrororvemaluetheand2leads.alsotlytotestaTheorembwettererythingpourerformanceand,ofthetheulltest.ypItwishareasonableetobuildconjectureparametricthatnatural:so,estimatorstestingfolloofwconsistencyTheatheforpusesrequiredmoreaecienceduretlyNotetheininformationbandterms.testingthevmoofdelwthan.,ethat(usedconditionsthe(a)-(e)situationyvbofreplacedfunction)doanesoinbutthisitmaseemsbdiculteasilytobderivaete-Carlosomedgeneralclosedpropareertiesatoailable.compareus,morethisaccuratelyork,simplyi.i.d.isAare,.satised..whereypIfisandfrom,nthenunder(6)b2.2toAbtesttermsforesp.elyThbus,thewfamilyeofc2.hoasymptoticosevidesrproconvergenceTheoremandThenInandsituation,one(5),immediatesemi-parametricTheWthiseithaisvthatasforantoestimator1,ofeab5outd (?)2
f(x;?)
? >0 ? >