Empirical Bayes and Likelihood Based Methods for Measurement Error Models

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Empirical Bayes and Likelihood Based Methods for Measurement Error Models
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Guo, Meixi
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Statistics
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Ghosh, Malay
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Rosalsky, Andrew J
Khare, Kshitij
Chen, Yunmei

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empirical-bayes -- james-stein-estimation -- measurement-error -- profile-likelihood -- small-area-estimation
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Abstract:
Measurement error is a problem that is commonly present in data collected from human respondents. Its early appearance dates back to nearly a century ago in econometrics. Measurement error in covariates then occurs in many other areas of statistics such as survey sampling, observational studies, survival data and longitudinal data. There is an enormous literature on this topic in linear regression. As is well known, measurement error in covariates causes biases in estimated regression coefficients. The primary task is thus to correct this bias. This dissertation concerns both estimation of regression coefficients under linear regression models and prediction of random effects under mixed effects linear models wherein the covariates are measured with error. Chapter 1 gives an overview of regression problems where covariates are subject to measurement error along with a simple illustrative example. A somewhat detailed summary of the subsequent chapters is also provided. Chapter 2-4 deal with independent topics; each chapter stands on its own ground. Inspired by Prasad and Rao’s (1993) rigorous second-order approximation of the MSE for a random effects model, Chapter 2 extends the work of Torabi, Datta and Rao (2009) who improved significantly on the EB estimators of Ghosh, Sinha and Kim (2006) by using information on both the covariate and response variable. We provide a second order correct (i.e., correct up to order O(m^(-1)), m being the sample size) estimator of the 8 MSE of the EB estimators in contrast to the first order correct (i.e., correct up to order O(m^(-1/2)), ) jackknife estimator of the MSE. Chapter 3 is related to Whittemore’s (1989) paper who showed how to rectify the inconsistency of the usual least squares estimator of the regression coefficient in a measurement error model by an alternative James-Stein (James and Stein, 1961) estimator. No MSE of her estimator was provided however. To fill this gap, we derive a second order correct estimator of the MSE of the Stein estimator under two cases: known measurement variance and unknown measurement variance. Chapter 4 of this dissertation introduces a new approach in dealing with measurement error problems. A likelihood-based method for mixed effect measurement error models is employed. We investigate maximum likelihood estimation under adjusted profile-likelihood. It is shown that the empirical predictors derived for random effects are first order asymptotically optimal in the sense of Robbins (1956). Chapter 5 provides a summary of our results, and provides a few pointers for future research.
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by Meixi Guo.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Ghosh, Malay.
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EMPIRICALBAYESANDLIKELIHOODBASEDMETHODSFORMEASUREMENTERRORMODELSByMEIXIGUOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012MeixiGuo 2

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Tomygrandmaandmyparents 3

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ACKNOWLEDGMENTS IhavebeendeeplyfortunatetoworkwithDr.MalayGhosh,whoseenthusiasmintheeldandhumblenessverymuchinuencedme,whoseencouragementandcoachinghavealwaysbeenacatalysttokeepmemovingforward.Ialsoexpressmygratitudetomycommitteemembersfortheirhelpfulcommentsandprecioustime.IthankDr.GeorgeCasella,Dr.Ghoshwhocametovisitmeduringmyillness,allthestuffs,facultiesfromthedepartmentandmyfriendsandfamilyfortheirpatienceandendlesssupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 10 2EMPIRICALBAYESESTIMATIONINRANDOMEFFECTSMODELSWITHCOVARIATESSUBJECTTOMEASUREMENTERROR ............. 14 2.1Background ................................... 14 2.2EmpiricalBayesPredictors .......................... 15 2.3MeanSquaredError .............................. 18 2.4Simulation .................................... 21 2.5SummaryandDiscussion ........................... 22 3MEANSQUAREDERROROFJAMES-STEINESTIMATORSFORMEASUREMENTERRORMODELS ........................ 23 3.1Background ................................... 23 3.2Themodelwithmeasurementvarianceknown ............... 24 3.2.1TheJames-Steinestimator ...................... 24 3.2.2MeanSquaredError .......................... 26 3.3Themodelwithmeasurementvarianceunknown .............. 32 3.4SimulationStudy ................................ 37 3.5Concludingremarks .............................. 39 4LIKELIHOODBASEDMETHODSFORSMALLAREAESTIMATIONWITHCOVARIATESSUBJECTTOMEASUREMENTERROR ............. 40 4.1Background ................................... 40 4.2Theprolelikelihoodapproach ........................ 41 4.3Themeansquarederror ............................ 50 4.4Simulation .................................... 52 4.5Summary .................................... 53 5SUMMARYANDCONCLUSIONS ......................... 55 APPENDIX ADERIVATIONOFTHEOREM2.3 .......................... 57 BDERIVATIONOFEQUATION3-24 ......................... 60 5

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BIBLIOGRAPHY ...................................... 61 BIOGRAPHICALSKETCH ................................ 65 6

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LISTOFTABLES Table page 2-1Simulationresults .................................. 22 3-1Performanceofvariousestimateswhenisknown ............... 37 3-2Performanceofvariousestimateswhenisestimated .............. 38 3-3Performanceoftheproposedestimateswithvarying .............. 38 4-1PerformanceoftheMSEofvariousestimates .................. 53 7

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEMPIRICALBAYESANDLIKELIHOODBASEDMETHODSFORMEASUREMENTERRORMODELSByMeixiGuoAugust2012Chair:MalayGhoshMajor:StatisticsMeasurementerrorisaproblemthatiscommonlypresentindatacollectedfromhumanrespondents.Itsearlyappearancedatesbacktonearlyacenturyagoineconometrics.Measurementerrorincovariatesthenoccursinmanyotherareasofstatisticssuchassurveysampling,observationalstudies,survivaldataandlongitudinaldata.Thereisanenormousliteratureonthistopicinlinearregression.Asiswellknown,measurementerrorincovariatescausesbiasesinestimatedregressioncoefcients.Theprimarytaskisthustocorrectthisbias.ThisDissertationconcernsbothestimationofregressioncoefcientsunderlinearregressionmodelsandpredictionofrandomeffectsundermixedeffectslinearmodelswhereinthecovariatesaremeasuredwitherror.Chapter1givesanoverviewofregressionproblemswherecovariatesaresubjecttomeasurementerroralongwithasimpleillustrativeexample.Asomewhatdetailedsummaryofthesubsequentchaptersisalsoprovided.Chapter2-4dealwithindependenttopics;eachchapterstandsonitsownground.InspiredbyPrasadandRao's(1993)rigoroussecond-orderapproximationoftheMSEforarandomeffectsmodel,Chapter2extendstheworkofTorabi,DattaandRao(2009)whoimprovedsignicantlyontheEBestimatorsofGhosh,SinhaandKim(2006)byusinginformationonboththecovariateandresponsevariable.Weprovideasecondordercorrect(i.e.,correctuptoorderm)]TJ /F6 7.97 Tf 6.59 0 Td[(1,mbeingthesamplesize)estimatorofthe 8

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MSEoftheEBestimatorsincontrasttotherstordercorrect(i.e.,correctuptoorderm)]TJ /F7 5.978 Tf 7.78 3.26 Td[(1 2)jackknifeestimatoroftheMSE.Chapter3isrelatedtoWhittemore's(1989)paperwhoshowedhowtorectifytheinconsistencyoftheusualleastsquaresestimatoroftheregressioncoefcientinameasurementerrormodelbyanalternativeJames-Stein(JamesandStein,1961)estimator.NoMSEofherestimatorwasprovidedhowever.Tollthisgap,wederiveasecondordercorrectestimatoroftheMSEoftheSteinestimatorundertwocases:knownmeasurementvarianceandunknownmeasurementvariance.Chapter4ofthisDissertationintroducesanewapproachindealingwithmeasurementerrorproblems.Alikelihood-basedmethodformixedeffectmeasurementerrormodelsisemployed.Weinvestigatemaximumlikelihoodestimationunderadjustedprole-likelihood.ItisshownthattheempiricalpredictorsderivedforrandomeffectsarerstorderasymptoticallyoptimalinthesenseofRobbins(1956).Chapter5providesasummaryofourresults,andprovidesafewpointersforfutureresearch. 9

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CHAPTER1INTRODUCTIONMeasurementerrororerrors-in-variables(R.J.Adcock,1877,1878;C.H.Kummell,1879;Koopmans,1937)isaproblemthatiscommonlypresentinmanyscienticareas.Itsearlyappearancedatesbacktonearlyacenturyagoineconometrics.Measurementerrorincovariatesoccursinmanyotherareasofstatisticssuchassurveysampling,observationalstudies,survivaldata,longitudinaldataandsoon.Oflate,ratherthanneglectingmeasurementerror,peoplehavestartedtodocumentandfacetheproblemsquarely.Often,attributabletothenatureofthedata,statisticiansarenotonlyinterestedinthetheoreticaldevelopmentofthemodel,butratherfocustheirattentiononthevalidityoftheassumption.SomerecentaccountincludesJudgeetal.(1985),Boundetal.(2001)andBiemeretal.(2003).Applicationshavethenbeenextendedtomanyotherdisciplines,suchasagronomy,socialsciences,medicineandenvironmentalepidemiology.Weciteafewexamples.I.SupposewewanttopredictthecornyieldsforseveralIowacounties,thecovariateusedhereissoilnitrogenlevel(Fuller,1987,p.2).Toestimatethenitrogenlevel,itisnecessarytosamplethesoilofanexperimentalplot,andtoperformalaboratoryanalysisonthesample.Asaresultofsamplingandlaboratoryanalysis,wedonotobservethetruenitrogenlevel,butonlyitsestimate.II.Supposewewanttostudywhetherobstructivesleepapnea(OSA)isassociatedwithcardiovascularmorbidity.TocomparetheOSAandnon-OSAgroups,measurementssuchasbloodpressure,heartrate,percentagebodyfatareneededfortheanalysis.However,manyofthesecovariatesarelikelytomeasuredwitherror,forexample,thestandardmanometerusedformeasuringbloodpressureinvolvingasleevewrappedroundthearminsteadofadirectmeasurementofintra-arterialbloodpressure.III.Supposeweareinterestedinestimatingthevolumeoftreesforseveralareas.Wehavedata(forsmallersub-areas)onvolumeofstemwoodforthecurrentyearand 10

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measuresofthediameterandheightofstemwoodfromapreviousyear.Here,wemaymodelthevolumeoftreesbytheirdiameterandheight.However,thelattertwoarelikelytobemeasuredwitherror.Oneofthekeyquestionsthatoccursinmeasurementerrorproblemsisthebiasthatmeasurementerrorintroducesintoparameterestimates.Webeginwithasimplelinearregressionmodel(Fuller,1987,p.3)yi=0+1xi+ei,i=1,2,...,m,wheretheerrorseiiidN)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(0,2e.Theusualunbiasedleastsquaresestimator(OLS)for1isthengivenby^1="mXi=1(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(x)2#)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(x)(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y).Nowassumethatwecannotobservethexi,butobserveinsteadXi,whereXi=xi+ui.WeassumethatthemeasurementerroruiiidN(0,),theunobservedvariablesxiiidN(,)aremutuallyindependentwiththeei.Asaresult,theOLSestimatesthatweactuallyobtainis~1="mXi=1)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(X2#)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(X(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y).If(Xi,yi)hasamultivariatenormaldistribution,wendthat E~1=1 +.(1)Heretheregressioncoefcientin( 1 )isattenuatedbythemeasurementerror(seeCarrolletal.2006,Figures3.1-3.2foranintuitiveillustration).Thesameideacanbeextendedtovectorcovariatesandmultivariatelinearmodels,aswaswellsummarizedinaclassicbookbyFuller(1987).Thebookdiscussedbothstructuralmeasurementerror(SME)modelsandfunctionalmeasurementerror 11

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(FME)modelsinatraditionalsense.InFMEmodels,unknowntruecovariatesarenon-stochastic,whereasinSMEmodelstheyaretreatedasrandom.Ontheotherhand,tomeettherapidlygrowingdemandonstatisticalmethodologiesparticularlywithinthelastthreedecades,Carrolletal.(2006)providedathoroughtreatiseofnonlinearmeasurementerrormodels.Thisgeneralizationenablesacleanerapproach,aswasaccountedbyCarrolletal,Italsoleadstomoreusefulmethodsofestimationandinferencethantheoldidea.InmyDissertation,Iwillconnemyattentiontolinearmodelscontainingcovariatemeasurementerror.BothFMEmodelsandSMEmodelsthatappearinlaterchapterswillbedenedintheclassicalsense.Chapter 2 considerslinearmixedmodelswithcovariatessubjecttomeasurementerror.Inthecontextofsmallareaestimation,Ghosh,SinhaandKim(2006)(GSK)andGhoshandSinha(2007)consideredsimultaneousestimationofstratameansanddevelopedempiricalBayes(EB)/hierarchicalBayes(HB)proceduresforFME/SMEmodels.GSKdemonstratedthattheirestimatorsweremuchsuperiortothedirectestimatorsofthestratameans.ThisworkextendedtheearlierworkofGhoshandMeeden(1986),GhoshandLahiri(1987,1992),PrasadandRao(1990)andDattaandGhosh(1991)whoconsideredEB/HBsmallareaestimation(predictingmixedeffects)whencovariateshadnomeasurementerror.Morerecently,Torabi,DattaandRao(2009)(TDR)improvedsignicantlyontheEBestimatorsofGSKbyusinginformationonboththecovariateandresponsevariable.However,theseauthorsusedthejackknifemethodtoobtainestimatorsofthemeansquarederror(MSE)oftheEBestimators.Theirapproachensuredarstordercorrect(i.e.,correctuptoO)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1=2,wheremisthesamplesize)MSEexpansion.Incontrasttotheirapproach,hereweprovideasecondordercorrect(i.e.,correctuptoO)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)estimatoroftheMSEoftheEBestimators.However,unlikeTorabietal.,weonlyconsiderbalanceddata,i.e.,wherethennumberofobservationspercellisxed. 12

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Forsomerecentworkonmixedmodelswithcovariatesmeasuredwitherror,werefertoPaternoandAmemiya(1996),Wang,Lin,GutierrezandCarroll(1998),etc.,formethodsongeneralsettingsandBuonaccorsietal.(2000),Tostesonetal.(1998)onlongitudinalsettingswithatime-varyingcovariatesubjecttomeasurementerror.Inarelatedbutdifferentlyfocusedpaper,Whittemore(1989)consideredestimationofregressioncoefcientsinclassicallinearregressionmeasurementerrormodels.ShehadtheveryinterestingresultwhichshowedhowtorectifytheinconsistencyoftheusualleastsquaresestimatoroftheregressionestimatorinameasurementerrormodelbyanalternativeJames-Stein(JamesandStein,1961)estimator.ItiswellknownthatStein'sestimatorisanEBestimator.Thus,Whittemore(1989)providedanimportantrationaleforapplyingEBmethodsinmeasurementerrorproblems.Shedidnotprovideanyrigorousprooffortheconsistencyofthesuggestedestimator.Also,noMSEofherestimatorwasprovided.InChapter 3 ,wedevelopasecondordercorrectestimatoroftheMSEoftheSteinestimatorundertwocases:knownmeasurementvarianceandunknownmeasurementvariance.Unliketheprecedingchapters,Chapter 4 investigatesthelikelihood-basedmethodsformixedeffectmeasurementerrormodelsinthesmallareacontext(Rao,2003).YbarraandLohr(2008)consideredaFay-Herriotmodelinthesmallareacontextwhereauxiliaryinformationissubjecttomeasurementerror.TheyappropriatelyadjustedthemethodofmomentapproachofPrasadandRao(1990)toestimatetheparameters.DattaandLahiri(2000)andDattaetal.(2005)showedthatforthearealevelmodelwithoutmeasurementerror,maximumlikelihood(ML)andresidualmaximumlikelihood(REML)estimatorsaremorestablethanthePrasad-Raoestimator.ThereforetoemploytheMLestimators,weproposetheprolelikelihoodapproach.Weremovethebiasinthescorefunctionsandobtainconsistentestimatorsoftheparameters.Empiricalpredictorsoftherandomeffectsarethenderived.ItisshownthattheEmpiricalpredictorsarerstorderasymptoticallyoptimalinthesenseofRobbins(1956). 13

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CHAPTER2EMPIRICALBAYESESTIMATIONINRANDOMEFFECTSMODELSWITHCOVARIATESSUBJECTTOMEASUREMENTERROR 2.1BackgroundRandomeffectsmodelshavebeenacornerstoneofsuccessinstatisticalmethodology.Itsapplicationextendstomanyotherdisciplines-animalscience,agronomy,economics,socialscience,epidemiology,medicine,genetics,justtonameafew.EmpiricalBayes(EB)methodshaveprovedtobequitevaluableinthiscontext,especiallywheninterestliesinsimultaneousestimationofseveralmixedeffects.Muchoftheliteratureinthisgeneralareaisbasedontheassumptionofnonstochasticcovariates.Situations,however,doarisewhencovariatesaresubjecttomeasurementerror.Averypracticalexampleinvolvesbloodpressuremeasurementsofseveralindividualsincertainxedpointsoftime.Thesecasescallforadifferentstatisticalanalysisthanwhatisusuallyappliedtotheusualrandomeffectsmodelswithnonstochasticcovariates.TheobjectiveofthischapteristoconsiderarandomeffectsmultipleregressionmodelwiththesamenumberofobservationspercellandobtainEBpredictorsofthemixedeffects.Whilesmallareaestimationproblemsalsoinvolvepredictionofmixedeffects,ourprocedureisnotparticularlyapplicabletosmallareaestimationsinceequalnumberofobservationspercellisnotveryrealisticthere.Rather,theresultsofthischapteraremoresuitableforproblemsinanimalscience,agronomyandotherswhereabalanceddesignisused(seeforinstance,Searle,1971,p.379;Khurietal,1998,p.39,p.47).WewouldliketomentionhowevertheworkofTorabi,DattaandRao(2009)whoconsideredasimplelinearnestederrorregressionmodelinthesmallareacontextandobtainedEBpredictorsofthemixedeffects.Theyobtainedmeansquarederrors(MSE)oftheseEBpredictorsusingajackknifeapproach.Incontrast,wendhereamorerenedestimatoroftheMSE.Moreover,weconsideramultiplelinearregressionmodel 14

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whichextendsthemodelofTorabietal.(2009).However,asmentionedalready,thepresentapproachisrestrictedtobalancedrandomeffectsmodels.Section 2.2 ofthischapterisdevelopmentofEBestimators.Section 2.3 developstheMSEoftheseEBestimators,andalsoestimatorsoftheseMSE's.Section 2.4 providessomesimulationresults.SomenalremarksaremadeinSection 2.5 2.2EmpiricalBayesPredictorsWebeginwiththenestederrorregressionmodel yij=0+xTi+ui+eij,(2) Xij=xi+ij,(2)wheretheui,eijandijaremutuallyindependentwitheijiidN)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(0,2e,ijiidN(0,)anduiiidN)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(0,2u(j=1,...,n;i=1,...,m).Also,boththeintercept0andtheregressionparameter(p1)areunknown.WecasttheprobleminaBayesianframeworkwith0+xTi+ui=isothatyijjiindN)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(i,2e,iindN)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(0+xTi,2u.Further,weconsiderastructuralregressionmodelwithxiiidN(,V),i=1,...,m.Theproblemisestimationoftheiundertheproposedstructuralmeasurementerrormodel.Itiseasytoseethatunderthegivenmodel,theminimalsufcientstatisticsis(yi,Xi,SSWy,SSWX),whereyi=n)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pnj=1yij,Xi=n)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pnj=1Xij(i=1,...,m),SSWy=Pmi=1Pnj=1(yij)]TJ /F9 11.955 Tf 13.14 0 Td[(yi)2andSSWX=Pmi=1Pnj=1)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Xij)]TJ /F9 11.955 Tf 14.44 2.66 Td[(Xi)]TJ /F4 11.955 Tf 12.95 -9.69 Td[(Xij)]TJ /F9 11.955 Tf 14.44 2.66 Td[(XiT.Conditionalonthexixed,undersquarederrorloss,theBayesestimatorofi(i=1,...,m)is ^Bi=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)yi+B)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(0+xTi,B=n)]TJ /F6 7.97 Tf 6.59 0 Td[(12e=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(n)]TJ /F6 7.97 Tf 6.58 0 Td[(12e+2u.(2)Werstpredictthexiby^xi=E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(xijyi,XiandobtainthepseudoBayespredictors ^PBi=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)yi+B)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(0+^xTi,i=1,...,m.(2) 15

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Tond^xi,westartwiththejointmultivariatenormaldistribution 0BBBB@yiXixi1CCCCAiidN2666640BBBB@0+T1CCCCA,0BBBB@!TVTVVn)]TJ /F6 7.97 Tf 6.59 0 Td[(1+VVVVV1CCCCA377775,i=1,...,m,(2)where!=2e=n+2u+TV.LetS=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1+V)]TJ /F6 7.97 Tf 6.59 0 Td[(1V,)-278(=V)]TJ /F4 11.955 Tf 11.87 0 Td[(VSandg=T(VS).Thefollowingtheoremndstheconditionaldistributionofxigiven(yi,Xi). Theorem2.1. xijyi,XiN(^xi,P),where^xi=+(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F9 11.955 Tf 5.76 -9.69 Td[(yi)]TJ /F3 11.955 Tf 11.95 0 Td[(0)]TJ /F3 11.955 Tf 11.95 0 Td[(T)]TJ /F3 11.955 Tf 6.77 0 Td[(+(ST)]TJ /F9 11.955 Tf 11.95 -.16 Td[((!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 6.77 0 Td[(TST))]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F3 11.955 Tf 11.96 0 Td[(, (2)P=)]TJ /F2 11.955 Tf 22.05 0 Td[()]TJ /F9 11.955 Tf 11.95 -.16 Td[((!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 6.77 0 Td[(T. (2) Proof. Fromthepropertiesofthemultivariatenormaldistribution, ^xi=+VV0B@!TVVn)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V1CA)]TJ /F6 7.97 Tf 6.58 0 Td[(10B@yi)]TJ /F3 11.955 Tf 11.96 0 Td[(0)]TJ /F3 11.955 Tf 11.96 0 Td[(TXi)]TJ /F3 11.955 Tf 11.96 0 Td[(1CA.(2)Byaformulaforinversionofpartitionalmatrices(e.g.,Rao,1973,p.33) 0B@!TVVn)]TJ /F6 7.97 Tf 6.58 0 Td[(1+V1CA)]TJ /F6 7.97 Tf 6.59 0 Td[(1=264(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F9 11.955 Tf 11.29 -.16 Td[((!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1TST)]TJ /F9 11.955 Tf 11.3 -.17 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1S)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1+V)]TJ /F3 11.955 Tf 11.95 0 Td[(!)]TJ /F6 7.97 Tf 6.59 0 Td[(1VTV)]TJ /F6 7.97 Tf 6.58 0 Td[(1375,(2)againbyastandardmatrixinversionformula(e.g.,Rao,1973,p.33))]TJ /F4 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V)]TJ /F3 11.955 Tf 11.96 0 Td[(!)]TJ /F6 7.97 Tf 6.58 0 Td[(1VTV)]TJ /F6 7.97 Tf 6.59 0 Td[(1=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V)]TJ /F6 7.97 Tf 6.58 0 Td[(1+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V)]TJ /F6 7.97 Tf 6.58 0 Td[(1VTV)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1+V)]TJ /F6 7.97 Tf 6.59 0 Td[(1=! 1)]TJ /F3 11.955 Tf 11.95 0 Td[(TV(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V))]TJ /F6 7.97 Tf 6.59 0 Td[(1V=!=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V)]TJ /F6 7.97 Tf 6.58 0 Td[(1+(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1STST. (2) 16

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Nowfrom( 2 )-( 2 ),^xi=+(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 6.78 0 Td[(ST+(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1VSTST)]TJ /F9 11.955 Tf 11.96 -.17 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1VTST (2)0B@yi)]TJ /F3 11.955 Tf 11.95 0 Td[(0)]TJ /F3 11.955 Tf 11.95 0 Td[(TXi)]TJ /F3 11.955 Tf 11.95 0 Td[(1CA=+(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 5.77 -9.68 Td[(yi)]TJ /F3 11.955 Tf 11.96 0 Td[(0)]TJ /F3 11.955 Tf 11.96 0 Td[(T)]TJ /F3 11.955 Tf 6.77 0 Td[(+(ST)]TJ /F9 11.955 Tf 11.96 -.17 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 6.78 0 Td[(TST))]TJ /F9 11.955 Tf 7.18 -7.03 Td[(Xi)]TJ /F3 11.955 Tf 11.95 0 Td[(. (2)Thisproves( 2 ). Next,frompropertiesofthemultivariatenormaldistribution,P=V)]TJ /F10 11.955 Tf 11.96 16.86 Td[(VV0B@!TVVn)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V1CA)]TJ /F6 7.97 Tf 6.58 0 Td[(10B@TVV1CA=V)]TJ /F9 11.955 Tf 11.96 -.17 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 6.78 0 Td[(TV)]TJ /F9 11.955 Tf 11.95 0 Td[((ST)]TJ /F9 11.955 Tf 11.95 -.17 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 6.78 0 Td[(TST)V=)]TJ /F2 11.955 Tf 18.73 0 Td[()]TJ /F9 11.955 Tf 11.95 -.17 Td[((!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 6.77 0 Td[(T. (2)Nextfrom( 2 )and( 2 ), ^PBi=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)yi+B[0+T+(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 5.77 -9.69 Td[(yi)]TJ /F3 11.955 Tf 11.96 0 Td[(0)]TJ /F3 11.955 Tf 11.96 0 Td[(T)]TJ /F3 11.955 Tf 12.96 -9.69 Td[(T)]TJ /F3 11.955 Tf 6.77 0 Td[(+)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F3 11.955 Tf 11.96 0 Td[(T(S)]TJ /F9 11.955 Tf 11.96 -.17 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(T)]TJ /F3 11.955 Tf 6.78 0 Td[(S)]. (2) Notingthat1)]TJ /F9 11.955 Tf 11.96 -.17 Td[((!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(T)]TJ /F3 11.955 Tf 6.77 0 Td[(=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=n+2u+T)]TJ /F3 11.955 Tf 6.77 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(T)]TJ /F3 11.955 Tf 6.77 0 Td[(=(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F6 7.97 Tf 6.59 0 Td[(1=AB)]TJ /F6 7.97 Tf 6.58 0 Td[(1,whereA=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=n(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1,itfollowsfrom( 2 )that ^PBi=yi)]TJ /F4 11.955 Tf 11.95 0 Td[(B)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(AB)]TJ /F6 7.97 Tf 6.59 0 Td[(1yi+B)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(AB)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 12.96 -9.68 Td[(0+T+A)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F3 11.955 Tf 11.95 0 Td[(TS=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(A)yi+Ah0+T+)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F3 11.955 Tf 11.96 0 Td[(TSi. (2) 17

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Noticethat^PBi=E)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ijyi,Xi.Therefore,thepseudoBayesestimatorobtainedherecanalsobevieweddirectlyasafullBayespredictor.NowtondtheEBestimatorofi,oneneedstoestimatealltheunknownquantitiesA,0,,andS.Deney=m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1yiandX=m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1Xi.SinceE)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(X=andE(y)=0+T,weestimatebyXand0+Tbyy.Further,deningMSWy=SSWy=(m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))andMSWX=SSWX=(m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)),itfollowsthatE(MSWy)=2eandE(MSWX)=sothatweestimate2eby^2e=MSWyandbyMSWX.Further,deneMSBy=mXi=1(yi)]TJ /F9 11.955 Tf 12.25 0 Td[(y)2=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1),MSBX=mXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(X)]TJ /F9 11.955 Tf 14.65 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(XT=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)andMSByX=mXi=1)]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(X(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y)=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1).NowfromE)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xijyi=+!)]TJ /F6 7.97 Tf 6.58 0 Td[(1V,itfollowsthatE(MSByX)=V.Moreover,E(MSBX)=n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V.Bythelawoflargenumbers,onecanestimateS=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V)]TJ /F6 7.97 Tf 6.59 0 Td[(1Vconsistentlyby(MSBX))]TJ /F6 7.97 Tf 6.59 0 Td[(1MSByX.Finally,notethatE(MSBy)=!sothatagainbythelawoflargenumbersg=T(VS)=(V)T)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+V)]TJ /F6 7.97 Tf 6.58 0 Td[(1VisconsistentlyestimatedbyMSBTyXMSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XMSByX.ThusweestimateA=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=n(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1by^A=n)]TJ /F6 7.97 Tf 6.59 0 Td[(1MSWy=)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(MSBy)]TJ /F4 11.955 Tf 11.95 0 Td[(MSBTyXMSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XMSByX.Hence,from( 2 ),onegetsanEBestimator ^EBi=1)]TJ /F9 11.955 Tf 12.85 2.66 Td[(^Ayi+^Ay+^A)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(XTMSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XMSByX.(2) 2.3MeanSquaredErrorWenowproceedtotheevaluationofmeansquarederroroftheEBestimators^EBi(i=1,...,m).WewilleventuallyprovideanapproximationtothisMSEwhichiscorrectuptoO)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Tothisend,werstprovethefollowingtheorem. Theorem2.2. E(^EBi)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2=E(^EBi)]TJ /F9 11.955 Tf 12.68 2.66 Td[(^PBi)2+E(^PBi)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2. Proof. Theproofrequiresshowing E(^EBi)]TJ /F9 11.955 Tf 12.68 2.66 Td[(^PBi)(^PBi)]TJ /F3 11.955 Tf 11.96 0 Td[(i)=0.(2) 18

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NotingthatEijyi,Xi=E[ijyi]=^Biandusingtheformulaforiteratedconditionalexpectations,thelefthandsideof( 2 )equalsE(^EBi)]TJ /F9 11.955 Tf 11.49 2.65 Td[(^PBi)(^PBi)]TJ /F9 11.955 Tf 11.49 2.65 Td[(^Bi).Now^PBi)]TJ /F9 11.955 Tf 11.49 2.65 Td[(^Bi=BT(^xi)]TJ /F4 11.955 Tf 11.96 0 Td[(xi)andExijyi,Xi=^xi.Hence, E(^EBi)]TJ /F9 11.955 Tf 12.69 2.65 Td[(^PBi)(^xi)]TJ /F4 11.955 Tf 11.95 0 Td[(xi)=Eh(^EBi)]TJ /F9 11.955 Tf 12.69 2.65 Td[(^PBi)E)]TJ /F9 11.955 Tf 5.66 -9.68 Td[(^xi)]TJ /F4 11.955 Tf 11.96 0 Td[(xijyi,Xii=0(2)andthus E(^EBi)]TJ /F9 11.955 Tf 12.69 2.65 Td[(^PBi)(^PBi)]TJ /F3 11.955 Tf 11.95 0 Td[(i)=E(^EBi)]TJ /F9 11.955 Tf 12.69 2.65 Td[(^PBi)(^PBi)]TJ /F9 11.955 Tf 12.68 2.65 Td[(^Bi)=0.(2)Thisprovesthetheorem. NextwendE(^PBi)]TJ /F3 11.955 Tf 11.95 0 Td[(i)2.Again,bythefactthatEijyi,Xi=^Bi,wegetE(^PBi)]TJ /F3 11.955 Tf 11.95 0 Td[(i)2=E(^PBi)]TJ /F9 11.955 Tf 12.69 2.66 Td[(^Bi)2+E(^Bi)]TJ /F3 11.955 Tf 11.95 0 Td[(i)2=B2TEh(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)Ti+2e n(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B). (2)ButEh(xi)]TJ /F9 11.955 Tf 12.13 0 Td[(^xi)(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)Ti=EEh(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)(xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)Tjyi,Xii=EV)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(xijyi,Xi=)]TJ /F2 11.955 Tf 22.05 0 Td[()]TJ /F9 11.955 Tf 11.96 -.16 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 6.78 0 Td[(T)]TJ ET BT /F1 11.955 Tf 433.45 -380.57 Td[((2)By( 2 )and( 2 ),E(^PBi)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=n(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)+B2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(T)]TJ /F3 11.955 Tf 6.78 0 Td[(1)]TJ /F9 11.955 Tf 11.95 -.17 Td[((!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F6 7.97 Tf 6.58 0 Td[(1T)]TJ /F3 11.955 Tf 6.77 0 Td[(. (2)Recallthat1)]TJ /F9 11.955 Tf 12.26 -.16 Td[((!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F6 7.97 Tf 6.59 0 Td[(1T)]TJ /F3 11.955 Tf 6.77 0 Td[(=AB)]TJ /F6 7.97 Tf 6.58 0 Td[(1sothatT)]TJ /F3 11.955 Tf 6.78 0 Td[(=(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g))]TJ /F9 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(AB)]TJ /F6 7.97 Tf 6.59 0 Td[(1=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=nA)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(AB)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Nowfrom( 2 ),E(^PBi)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=nB)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(AB)]TJ /F6 7.97 Tf 6.59 0 Td[(1=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n(1)]TJ /F4 11.955 Tf 11.95 0 Td[(A). (2)Nextweprovetheresult Theorem2.3. E(^EBi)]TJ /F9 11.955 Tf 12.68 2.66 Td[(^PBi)2=m)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=nA)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(p+3+2(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1. 19

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TheproofoftheTheoremislongandtechnical,andisdeferredtotheAppendix.InviewofTheorem 2.2 ,( 2 )andTheorem 2.3 ,onegetstheMSEapproximationresult E(^EBi)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n1)]TJ /F4 11.955 Tf 11.96 0 Td[(A+Am)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(p+3+2(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(2)Remark1.Inthespecialcasewhenp=1(i.e.forsimplelinearregression),onegetsE(^EBi)]TJ /F3 11.955 Tf 11.95 0 Td[(i)2=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n1)]TJ /F4 11.955 Tf 11.96 0 Td[(A+2Am)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(2+(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1.WenowderiveasecondorderunbiasedestimatoroftheMSEof^EBi,namelyanestimatorwhichiscorrectuptoO)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i.e.,theresidualtermiso)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Withtheestimator^2e=MSWyfor2e,bytheindependenceofMSWywith(MSBy,MSBX,MSByX),andthefactthat(m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))^2e2e2m(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1), E^2e=n1)]TJ /F9 11.955 Tf 12.85 2.66 Td[(^A=E)]TJ /F9 11.955 Tf 5.88 -9.68 Td[(^2e=n)]TJ /F4 11.955 Tf 11.95 0 Td[(E)]TJ /F9 11.955 Tf 5.89 -9.68 Td[(^2e=n2E(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g))]TJ /F6 7.97 Tf 6.59 0 Td[(1=2e=n)]TJ /F10 11.955 Tf 11.95 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n21+2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1E(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g))]TJ /F6 7.97 Tf 6.58 0 Td[(1. (2) InordertondE(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g))]TJ /F6 7.97 Tf 6.58 0 Td[(1,weneedthefollowinglemma Lemma1. (m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g)2m)]TJ /F8 7.97 Tf 6.58 0 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Proof. Werstwrite(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)=mXi=1(yi)]TJ /F9 11.955 Tf 12.25 0 Td[(y)2)]TJ /F10 11.955 Tf 11.96 20.45 Td[("mXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTyi#"mXi=1)]TJ /F9 11.955 Tf 7.18 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT#)]TJ /F6 7.97 Tf 6.59 0 Td[(1"mXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(Xyi#=uTGu,whereG=Im)]TJ /F4 11.955 Tf 12.97 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Jm)]TJ /F10 11.955 Tf 12.97 38.38 Td[(266664)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(X1)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT...)]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xm)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT377775SSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XX1)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XXm)]TJ /F9 11.955 Tf 13.64 2.65 Td[(XanduT=(y1...ym).ItiseasytocheckthatGissymmetricandidempotent.Thusrank(G)=tr(G)=m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(p.Further,conditionalonX1,...,Xm, uNE)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(ujX1,...,Xm,(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)Im,(2) 20

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whereE)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(ujX1,...,Xm=0+T+)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(X)]TJ /F3 11.955 Tf 11.95 0 Td[(TS1m+266664)]TJ /F9 11.955 Tf 8.85 -7.02 Td[(X1)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTS...)]TJ /F9 11.955 Tf 10.14 -7.03 Td[(Xm)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTS377775.Also,GE)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(ujX1,...,Xm=266664)]TJ /F9 11.955 Tf 8.84 -7.03 Td[(X1)]TJ /F9 11.955 Tf 13.64 2.65 Td[(XTS...)]TJ /F9 11.955 Tf 10.14 -7.03 Td[(Xm)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTS377775)]TJ /F10 11.955 Tf 11.96 38.38 Td[(266664)]TJ /F9 11.955 Tf 8.84 -7.03 Td[(X1)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XTS...)]TJ /F9 11.955 Tf 10.14 -7.03 Td[(Xm)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTS377775=0.Thus,uTGu=u)]TJ /F4 11.955 Tf 11.95 0 Td[(E)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(ujX1,...,XmTGu)]TJ /F4 11.955 Tf 11.95 0 Td[(E)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(ujX1,...,Xm.FromthesymmetryandidempotencyofG,itfollowsnowthatconditionalonX1,...,Xm,uTGu(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g)2m)]TJ /F8 7.97 Tf 6.59 0 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1.SincethisconditionaldistributiondoesnotdependonX1,...,Xm,itfollowsthatevenunconditionally(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)=uTGu(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g)2m)]TJ /F8 7.97 Tf 6.58 0 Td[(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1. Nowfrom( 2 )andLemma 1 E^2e=n1)]TJ /F9 11.955 Tf 12.85 2.66 Td[(^A=2e=n)]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=n21+2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1E(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g))]TJ /F6 7.97 Tf 6.59 0 Td[(1=2e=n)]TJ /F10 11.955 Tf 11.96 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n21+2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1m)]TJ /F9 11.955 Tf 11.96 0 Td[(1 m)]TJ /F4 11.955 Tf 11.95 0 Td[(p)]TJ /F9 11.955 Tf 11.96 0 Td[(3(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=n(1)]TJ /F4 11.955 Tf 11.96 0 Td[(A))]TJ /F4 11.955 Tf 11.95 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2e=nA[p+2+2(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1]+O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(2. (2) Similarly,E^A=A+O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Hence,from( 2 )and( 2 ),E(^EBi)]TJ /F3 11.955 Tf 12.96 0 Td[(i)2isestimatedby )]TJ /F9 11.955 Tf 5.89 -9.69 Td[(^2e=nh1)]TJ /F9 11.955 Tf 12.85 2.66 Td[(^A+^Am)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(2p+5+4(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1))]TJ /F6 7.97 Tf 6.58 0 Td[(1i.(2) 2.4SimulationWenowillustrateourmethodsusingasimplesimulationexperiment.Considerthecaseinwhichp=2,m=50,n=4,o=1,=(3,2)T,eijiidN(0,9),uiiidN(0,4),ijiidN0B@0,0B@42241CA1CAandxiiidN0B@0,0B@45581CA1CA.ThechoiceofthevarianceandcovariancematricesaretakenfromasimulationstudygiveninYbarraandLohr 21

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Table2-1. Simulationresults EmpiricalMeanSquaredErrorAnalyticalMeanSquaredError yi^EBi,na^EBi^EBi2.252.182.042.042.232.162.032.042.292.222.072.05 (2008)thatalsodealingwiththeprobleminvolvingmeasurementerror.Wecomparedourproposedestimator^EBitothedirectestimatoryiandthenaiveempiricalBayesestimator^EBi,naassumingtheobservedcovariatesXijarethetruevalues(supposeXijisxed).Table 2-1 reportstheaverageempiricalmeansquarederrorsfortheareascalculatedbasedonthousanditerations.Theaveragesoftheunbiasedestimatorsofthemeansquarederrorsbasedon( 2 )arealsogiveninthetable.Notsurprisingly,thedirectestimatorhasthelargestmeansquarederroramongtheestimatorsconsidered.Further,ourproposedestimator^EBiimprovesontheempiricalBayesestimator^EBi,nawithrespecttoefciencysincethelatterdoesnotincorporatemeasurementerror.Table 2-1 alsoprovidesstrongevidencetotheunbiasedestimateobtainedin( 2 );theanalyticalestimatesarealmostidenticaltotheempiricalresults. 2.5SummaryandDiscussionThischapterconsidersanestederrorlinearregressionmodelwithmultiplecovariatessubjecttomeasurementerror.Acorrectsecond-orderunbiasedapproximationtotheMSEoftheEBpredictorisobtained.However,intheunbalancedcasewherenvarieswitheachi,theexchangeabilitywhichhasbeenrepeatedlyappliedintheproofsofthetheoremswillnotholdanymore,therebycausingadditionaldifcultiesinthecalculations. 22

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CHAPTER3MEANSQUAREDERROROFJAMES-STEINESTIMATORSFORMEASUREMENTERRORMODELS 3.1BackgroundStein'sruleisgenerallyrecognizedasanimportanttoolforyieldingbiasedbutefcientestimators.Itdominatesthemaximumlikelihoodestimator(MLE)byprovidinguniformlylowertotalrisk.SinceStein'srulearisesnaturallyinanempiricalBayessetting,itenjoysonegoodpropertyofempiricalBayesestimators,namely,incontrasttotheBayesrule,frequentistriskaswellastheBayesriskoftheJames-Steinestimatorisrobustagainstmisspecifyingthepriorinformation.Itcanbeshownhowever,thatStein'sruleisdominatedbyitspositivepart(seee.g.,LehmannandCasella,p.276).Thisisbecauseonlythepositivepartactuallysucceedsinshrinkingtheestimatortowarditsshrinkingtargetalwaysintherightdirection.WhiletheSteinruleisofinterestbyitselfinmanyapplications,Whittemore(1989)inaninterestingpaperconsideredestimationofregressioncoefcientsinmeasurementerrormodels.ShehadtheveryinterestingresultwhichshowedhowtoovercometheinconsistencyoftheusualleastsquaresestimatoroftheregressionparameterinameasurementerrormodelbyanalternativeJames-Stein(JamesandStein,1961)estimator.Althoughtheauthordidnotprovidearigorousprooffortheconsistencyofthesuggestedestimator,orderiveditsMSE,theideapresentedinthepapermotivatedanadditionalgoodpropertyfortheJames-Steinestimatorunderthemeasurementerrorsetting.Simpleconsistentestimatorshavebeenconstructedfortheregressionparametersforthecaseofknownmeasurementvariance(Theil,1971,Johnston,1963andSchneeweiss,1976).OurobjectiveistoprovidearigoroussecondorderexpansionofthemeansquarederroroftheproposedJames-Steinestimatorbothunderknownmeasurementvarianceandunknownmeasurementvariance.Section 3.2 givesabriefreviewofWhittemore's(1989)paper,andalsoderivesthemeansquarederrorwhenthemeasurementvarianceisknown.Section 3.3 develops 23

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themeansquarederrorwhenthemeasurementvarianceisunknown.AnumericalexperimentisconductedinSection 3.4 .Section 3.5 providessomenalremarks. 3.2Themodelwithmeasurementvarianceknown 3.2.1TheJames-SteinestimatorWestartwiththeclassicallinearregressionmodelwithregressioncoefcientsmeasuredwitherror.Themodelisgivenby yi=0+1xi+ei,(3) zi=xi+ui,(3)wheretheresidualtermeiandthemeasurementerroruiareassumedtobeindependentwitheiiidN)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(0,2eanduiiidN(0,)(i=1,...,m).Also,supposethatthetruecovariatesxiarei.i.d.fromaGaussiandistributionxiiidN(,)andareindependentwith(ei,ui).Herebothandareunknown.Inthissection,wewillconsiderthecasewhenmeasurementvarianceisknown,whereasthenextsectionwillconsiderthecasewhenisunknown.Theobjectivewillalwaysbefocusingonestimationoftheregressionparameter1.Further,iftheresidualvariance2eisunknown,itcanbeeasilyestimatedusingrepeatedmeasurementsofyi.Weassumeinthissectionthat2eisknown.Asisgenerallyknown,byplugginginthenaiveestimatorziofxiintoaM-equation mXi=1(yi,xi,)=0,(3)onewouldgetaninconsistentestimatorfortheregressionparametersinthepresenceofmeasurementerror.Hereforourmodel(y,x,)isthescorefunctionsuchthat(y,x,)=(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)x.Whittemore(1989)suggestedasimpleeasy-to-applyapproachforobtainingconsistentestimatorsfortheregressionparameters:replacingthetruecovariatesxibytheirJames-Stein(J-S)estimators,andthensolvingEquation 3 inthe 24

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usualway.Asusual,inthischapterweadoptsquarederrorlossfunction.Wewillderiveacorrectsecondorderexpansionforthemeansquarederroroftheestimator.Supposem>3.FollowingWhittemore(1989),werstobtaintheBayesestimatorofxi: ^xi=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)zi+B,(3)wheretheshrinkagefactorB==(+).Now,togettheJ-Sestimator,oneneedstoprovideanunbiasedestimatoroftheunknownparametersandB.SincemarginallyziiidN(,+),^=zisanunbiasedestimatorofand^B=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=Pmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2isanunbiasedestimatorofB(EfronandMorris(1975)).TheJ-Sestimatorofxiisthengivenby ^xiJS=1)]TJ /F9 11.955 Tf 13.41 2.66 Td[(^Bzi+^Bz.(3)WepursueastepfurtherforthepositivepartSteinestimatorsincethelatterhaslowerorequalriskthantheJ-Sestimatorforeveryxi ^xiJS+=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^Bzi+^^Bz,(3)where^^B=min^B,(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1).Fortechnicalreasons,wedonottake^^Bastheminimumof^Band1.However,withlargem,theconstant(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)tendsto1.Let^xJS+=m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1^xiJS+.Thesuggestedconsistentestimatorof1,basedontheobservedcovariatesziis^^1=mXi=1yi)]TJ /F9 11.955 Tf 5.66 -9.68 Td[(^xiJS+)]TJ /F9 11.955 Tf 12.14 1.69 Td[(^xJS+=mXi=1)]TJ /F9 11.955 Tf 5.66 -9.68 Td[(^xiJS+)]TJ /F9 11.955 Tf 12.14 1.69 Td[(^xJS+2=mXi=1yi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)="1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^BmXi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2#. (3)Wewillshowinthenextsectionthat^^1isaconsistentestimatorof1. 25

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3.2.2MeanSquaredErrorInthissectionwewillderiveasecond-orderMSEapproximationtotheJ-Sestimates^^1under( 3 )and( 3 ).Inordertondtheexpansion,webeginwithafewlemmas. Lemma2. g(x)=g(a)+(x)]TJ /F4 11.955 Tf 11.96 0 Td[(a)g(1)(a)+1 2(x)]TJ /F4 11.955 Tf 11.96 0 Td[(a)2g(2)(a)++1 n!(x)]TJ /F4 11.955 Tf 11.95 0 Td[(a)ng(n)(a)+Rn(x,a),wheretheremainderterm Rn(x,a)=1 n!(x)]TJ /F4 11.955 Tf 11.95 0 Td[(a)n+1Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()ng(n+1)[x+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()a]d.(3) Proof. WereferthereadertoApostol(1967,Sec.7.5)foraproof. ThelemmaprovidesanicealternativeintegralexpressiontotheremaindertermoftheTaylorseries.Itwillbelaterusefulinndingthecorrectorderoftheremainderterminthemeansquarederrorexpansion. Lemma3. Supposez1,...,zmarei.i.dwithE(z1)=andEjz1j2r<1,wherer1.ThenEjz)]TJ /F3 11.955 Tf 11.95 0 Td[(j2r=O(m)]TJ /F8 7.97 Tf 6.58 0 Td[(r),forallr1. Proof. SeeSering(1980,lemmaB,p.68). UsingLemma 3 ,onecaneasilyprovethenextlemmawhichwillbeusedrepeatedlyinthesequel. Lemma4. P^B>(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=O(m)]TJ /F8 7.97 Tf 6.58 0 Td[(r),forallr1. Proof. Forallr1,let^B=B(m)]TJ /F6 7.97 Tf 6.59 0 Td[(3) Q,whereQ=Pmi=1(zi)]TJ /F9 11.955 Tf 12.11 0 Td[(z)2=(+)2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.ThenP^B>(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=P1 ^B<(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=PQ B(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)<(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=PQ m)]TJ /F9 11.955 Tf 11.96 0 Td[(11)]TJ /F4 11.955 Tf 11.96 0 Td[(BPj1)]TJ /F4 11.955 Tf 23.45 8.09 Td[(Q m)]TJ /F9 11.955 Tf 11.96 0 Td[(1j>1)]TJ /F4 11.955 Tf 11.96 0 Td[(BE)]TJ /F8 7.97 Tf 12.09 -4.98 Td[(Q m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(12r (1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2r, 26

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whereatthelaststep,weusedtheMarkov'sinequality.SinceQfollowsachi-squaredistributionwithm)]TJ /F9 11.955 Tf 12.32 0 Td[(1degreesoffreedom,itcanbeexpressedasthesumofsquaresofm)]TJ /F9 11.955 Tf 12.25 0 Td[(1independent,standardnormalrandomvariables,say,Q=Pm)]TJ /F6 7.97 Tf 6.59 0 Td[(1j=1Zj,whereZjarei.i.d.21variables.ThereforeitfollowsfromLemma 3 thatE(Q=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F9 11.955 Tf 11.95 0 Td[(1)2r=EhPm)]TJ /F6 7.97 Tf 6.59 0 Td[(1j=1Zj=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F9 11.955 Tf 11.95 0 Td[(1i2r=O(m)]TJ /F8 7.97 Tf 6.58 0 Td[(r).Hence,P^B>(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)=O(m)]TJ /F8 7.97 Tf 6.59 0 Td[(r). Lemma 4 showsinparticularthattheasymptoticcalculationswillbefacilitatedwhen^Bissubstitutedfor^^Bintheexpressionsinvolving^^B.Thiswillbeveryusefulinprovingthemaintheoremofthissection. Theorem3.1. Under( 3 )and( 3 ), E^^1)]TJ /F3 11.955 Tf 11.95 0 Td[(12=2e=+21(1+B)B=m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(3) Proof. Webeginwiththeequation E^^1)]TJ /F3 11.955 Tf 11.96 0 Td[(12=V^^1+hE^^1)]TJ /F3 11.955 Tf 11.96 0 Td[(1i2,(3)sincemarginally0B@yizi1CAiidN2640B@0+11CA,0B@2e+2111+1CA375,wehaveyijziindN0+1+1(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)(zi)]TJ /F3 11.955 Tf 11.95 0 Td[(),2e+21(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B).Thus,E^^1=EhE^^1jz1,...,zni=E24E0@Pmi=1yi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z) 1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^BPmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2jz1,...,zn1A35=1E 1)]TJ /F4 11.955 Tf 11.95 0 Td[(B 1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B!, 27

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sothat hE^^1)]TJ /F3 11.955 Tf 11.96 0 Td[(1i2=21"E 1)]TJ /F4 11.955 Tf 11.95 0 Td[(B 1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B)]TJ /F9 11.955 Tf 11.95 0 Td[(1!#2.(3)Also, V^^1=VhE^^1jz1,...,zni+EhV^^1jz1,...,zni=V"11)]TJ /F4 11.955 Tf 11.96 0 Td[(B 1)]TJ /F9 11.955 Tf 13.42 5.31 Td[(^^B#+E2642e+21(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B) 1)]TJ /F9 11.955 Tf 13.42 5.31 Td[(^^B2Pmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2375. (3) Tocomputetheseexpectations,letg1(x)=1=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x).ThenapplyingLemma 2 tox=^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B),centeredatzeroandn=2,onegets(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)=1)]TJ /F9 11.955 Tf 13.42 5.31 Td[(^^B=1+^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)+^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+1 2^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B3 (1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)3Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2g(3)1h^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)id.Nowtakingtheexpectationonbothsides,itfollowsthat Eh(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^Bi=1+E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)+E^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+Ef^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B3Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2g(3)1h^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)idg=2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)3 (3) Notethatsince^B=B(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=Q,whereQ=Pmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2=(+)2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,E^B=BandV^B=2B2=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(5).Hence,thesecondtermontheright-handsideof( 3 )isE^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=E^B)]TJ /F4 11.955 Tf 11.95 0 Td[(BI[^Bm)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]+m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(BI[^B>m)]TJ /F7 5.978 Tf 5.75 0 Td[(3 m)]TJ /F7 5.978 Tf 5.75 0 Td[(1]=E^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B)]TJ /F4 11.955 Tf 11.96 0 Td[(E^B)]TJ /F4 11.955 Tf 11.95 0 Td[(BI[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]+m)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(BI[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]=)]TJ /F4 11.955 Tf 9.3 0 Td[(E^B+m)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(2BI[^B>m)]TJ /F7 5.978 Tf 5.75 0 Td[(3 m)]TJ /F7 5.978 Tf 5.75 0 Td[(1]. 28

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BytheCauchy-Schwarzinequality,E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=E^B+m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(2BI[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]"E^B+m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(2B2#1 2P1 2^B>(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1),=O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F8 7.97 Tf 6.58 0 Td[(rforallr1byLemma 4 andthatE^B+m)]TJ /F6 7.97 Tf 6.59 0 Td[(3 m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(2B2isbounded.Similarly,E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B2=V^B)]TJ /F4 11.955 Tf 11.96 0 Td[(E"^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B2I[^B>m)]TJ /F7 5.978 Tf 5.75 0 Td[(3 m)]TJ /F7 5.978 Tf 5.75 0 Td[(1]+m)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B2I[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]#=2B2 m+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Nowweprovethefollowinglemmawhichshowsthatthelasttermontherighthandsideof( 3 )iso)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1. Lemma5. Under( 3 )and( 3 ),E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B3Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2g(3)1h^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)id=o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Proof. WestartwiththeintegralZ10(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2g(3)1h^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)id=3Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)4d.Let"=1 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B).Thenforlargem,""id 29

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Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2d+Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()21)]TJ /F9 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F6 7.97 Tf 6.58 0 Td[(4d+Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()21)]TJ /F3 11.955 Tf 11.96 0 Td[(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B))]TJ /F6 7.97 Tf 6.58 0 Td[(4Ih^^B)]TJ /F8 7.97 Tf 6.58 0 Td[(B>"id1 3+1 31)]TJ /F9 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F6 7.97 Tf 6.59 0 Td[(4+1 31)]TJ /F10 11.955 Tf 11.95 16.86 Td[(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B))]TJ /F6 7.97 Tf 6.58 0 Td[(4Ih^^B)]TJ /F8 7.97 Tf 6.58 0 Td[(B>"i=1 3+16 3+1 3((1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=2)4Ih^^B)]TJ /F8 7.97 Tf 6.58 0 Td[(B>"i.Hence,E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B3Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2g(3)1h^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)idE^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B31 3+16 3+1 3((1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)=2)4Ih^^B)]TJ /F8 7.97 Tf 6.59 0 Td[(B>"i17 3E^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B61 2+1 3((1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)=2)4E^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B61 2P^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B>"usingtheCauchy-Schwarzinequality.Withtediousbutstraightforwardcalculations,wegetE^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B6=E^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B6+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(3=O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(3byLemma 4 since[B(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)]=^B2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thus,forr1,P^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B>"P^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B>"P^Bm)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1+Pm)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B>"P^B>m)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1E264^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B2r "2r375+P^B>m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1=O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F8 7.97 Tf 6.58 0 Td[(rbyLemma 4 andE^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B2r=O(m)]TJ /F8 7.97 Tf 6.58 0 Td[(r).Hence,E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B3Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2g(3)1h^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)id=o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1whichprovesthelemma. Thus, Eh(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)=1)]TJ /F9 11.955 Tf 13.42 5.31 Td[(^^Bi=1+2B2=m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,(3) 30

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andhence hE^^1)]TJ /F3 11.955 Tf 11.96 0 Td[(1i2=421B4=m2(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)4+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(2.(3)Nowletg2(x)=1=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x)2,applyingLemma 2 againtox=^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B),oneobtains E(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2=1)]TJ /F9 11.955 Tf 13.42 5.31 Td[(^^B2=1+6B2=m(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1.(3)TheproofissimilartothederivationofEquation 3 ,andisomitted.Hence, V 1)]TJ /F4 11.955 Tf 11.96 0 Td[(B 1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^B!=E(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2=1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^B2)]TJ /F10 11.955 Tf 10.2 20.45 Td[("E 1)]TJ /F4 11.955 Tf 11.96 0 Td[(B 1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^B!#2=2B2=m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(3)Now,E"1=1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B2mXi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2#=E"1=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B2mXi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2nI[^Bm)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]+I[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]o#=E^^B=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B2I[^Bm)]TJ /F7 5.978 Tf 5.75 0 Td[(3 m)]TJ /F7 5.978 Tf 5.75 0 Td[(1]+E"2 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F6 7.97 Tf 6.58 0 Td[(2=mXi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2I[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]#=E^^B=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B2)]TJ /F4 11.955 Tf 11.95 0 Td[(E^^B=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)1)]TJ /F9 11.955 Tf 13.42 5.31 Td[(^^B2I[^B>m)]TJ /F7 5.978 Tf 5.76 0 Td[(3 m)]TJ /F7 5.978 Tf 5.76 0 Td[(1]+E"2 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)]TJ /F6 7.97 Tf 6.59 0 Td[(2=mXi=1(zi)]TJ /F9 11.955 Tf 12.11 0 Td[(z)2I[^B>m)]TJ /F7 5.978 Tf 5.75 0 Td[(3 m)]TJ /F7 5.978 Tf 5.75 0 Td[(1]#=1 (m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)E1=1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^B2)]TJ /F9 11.955 Tf 11.95 0 Td[(1=1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B)]TJ /F4 11.955 Tf 13.15 8.09 Td[(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1 4P^B>m)]TJ /F9 11.955 Tf 11.96 0 Td[(3 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1=B m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,wherethelasttwoequalitiesfollowfromLemma 4 ,theCauchy-Schwarzinequality,thefactthatPmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2(+)2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,( 3 )and( 3 ).Alongwith( 3 )and( 3 ),onegetsV^^1=221B2=m(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2+2e+21(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)B=m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 -.01 Td[(1.Thisprovesthetheorem. 31

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Remark2.ItfollowsfromTheorem 3.1 thatE^^1)]TJ /F3 11.955 Tf 11.95 0 Td[(12!0asm!1forall1.Hence,byMarkov'sinequality^^1isconsistentforestimating1.Ontheotherhand,fortheinconsistentestimator~1=Pmi=1yi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)=Pmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2,itiseasytoverifythatE~1)]TJ /F3 11.955 Tf 11.96 0 Td[(12=21B2+2e=+21(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)B=m,whichinvolvesalsotheO(1)ordertermduetothebiascausedbytheestimator.Remark3.ItisinterestingtonotethattheasymptoticvarianceofFuller'sintuitiveestimator(Fuller,1987,p.14)underthestructuralmeasurementmodelagreeswiththersttermontherighthandsideof( 3 ).However,arigorousjusticationoftheMSEthenrequiresauniformintegrabilityargument.WehaveinsteadprovidedadirectrigorouscalculationoftheMSEoftheestimatorgivenbyWhittemore.Remark4.AccordingtoFuller(Fuller,1987,p.17),thenormalityassumptionfortheerrortermscanberelaxedtosomemodestextentifoneassumesthattheerrorvariancesforeianduiaresmallrelativetothoseofthetruecovariatesfortheirsimpleestimator.ThusonthebasisofRemark3,oneestablishesrobustnessoftheasymptoticdistributionoftheestimatorof1asconsideredhere.WecannowprovideanestimatortotheMSEapproximationgivenin( 3 )bydirectlyplugginginconsistentestimatesfor1andB,namely^^1and^B(or^^B). 3.3ThemodelwithmeasurementvarianceunknownThissectionexaminesthecasewhenthevarianceofthemeasurementerrorisunknownandneedstobeestimated.Onceagain,westudyestimationofthemeansquarederror.Notsurprisingly,thetheoremsandlemmasneededinthepresentsectionarelargelythesameastheonesintheprecedingsection,theonlydifferencecomingintheproofofLemma 4 wherethereisanadditionalrandomfactorforestimatingtheunknownmeasurementvariance.Proofsidenticalorsimilartotheonesintheprevioussectionareomittedinthissection.Inaddition,wewillretaintheearliernotations.Forinstance,the 32

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shrinkagefactorinthepresentsectionhasthesamenotationB,butreferstoadifferentexpression.Toestimatetheunknownmeasurementvariance,weneedrepeatedmeasurementsontheobservedvalues.Wealsoassumethat2eisunknown,therefore,themodelbecomes yij=0+1xi+eij,(3) zij=xi+uij,(3)whereeij,uijandxi(i=1,...,m;j=1,...,n)aremutuallyindependentwitheijiidN)]TJ /F9 11.955 Tf 5.47 -9.68 Td[(0,2e,uijiidN(0,)andxiiidN(,).NowtheBayesestimatorforxiis ^xi=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)zi+B,(3)wherezi=n)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pjzij,B=(=n)=(=n+).WenowconsidertheunbiasedestimatorofBasgivenbyEfronandMorris(1972,sec.7):^B=n)]TJ /F6 7.97 Tf 6.58 0 Td[(1^(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=Pi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2,where^=Pi,j(zij)]TJ /F9 11.955 Tf 12.1 0 Td[(zi)2=[m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+2],z=(mn))]TJ /F6 7.97 Tf 6.59 0 Td[(1Pi,jzij.Further,oneobtainsthepositivepartSteinestimator^^B=min^B,(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1).Again,weestimatebyz,sothat^xiJS+=1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^Bzi+^^Bz,sothattheSteinestimatorof1is^^1=mXi=1yi)]TJ /F9 11.955 Tf 5.66 -9.68 Td[(^xiJS+)]TJ /F9 11.955 Tf 12.14 1.69 Td[(^xJS+=mXi=1)]TJ /F9 11.955 Tf 5.66 -9.68 Td[(^xiJS+)]TJ /F9 11.955 Tf 12.14 1.69 Td[(^xJS+2=mXi=1yi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)="1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^BmXi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2#. (3)Now,marginally0B@yizi1CAiidN2640B@0+11CA,0B@2e=n+2111=n+1CA375.Hence, yijziindN0+1+1(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)(zi)]TJ /F3 11.955 Tf 11.96 0 Td[(),2e=n+21(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)=n.(3) 33

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Nowbytheiteratedexpectationsformula, E^^1)]TJ /F3 11.955 Tf 11.95 0 Td[(12=V^^1+hE^^1)]TJ /F3 11.955 Tf 11.96 0 Td[(1i2=VhE^^1jz1,...,zni+EhV^^1jz1,...,zni+nEhE^^1jz1,...,zni)]TJ /F3 11.955 Tf 11.96 0 Td[(1o2=V"11)]TJ /F4 11.955 Tf 11.95 0 Td[(B 1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B#+E2642e=n+21(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)=n 1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B2Pmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2375+21"E 1)]TJ /F4 11.955 Tf 11.95 0 Td[(B 1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B)]TJ /F9 11.955 Tf 11.96 0 Td[(1!#2. (3) wherethelastequalityfollowsfromtheconditionaldistributiongivenin( 3 ).AnalogoustoLemma 4 inSection 3.2.2 ,wenowprovethefollowinglemma. Lemma6. P^B>(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=O(m)]TJ /F8 7.97 Tf 6.58 0 Td[(r),forallr1. Proof. LetA=Pin(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1),thenA[n=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)](=n+)2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Inaddition,[m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)+2]^=Pi,j(zij)]TJ /F9 11.955 Tf 12.1 0 Td[(zi)22m(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1),andisdistributedindependentlyofA.Thusforallr1,P^B>(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=P (m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)^=Xin(zi)]TJ /F9 11.955 Tf 12.11 0 Td[(z)2>1!=P ^ A)]TJ /F4 11.955 Tf 11.95 0 Td[(B>1)]TJ /F4 11.955 Tf 11.96 0 Td[(B!E^ A)]TJ /F4 11.955 Tf 11.96 0 Td[(B2r (1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2r=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B))]TJ /F6 7.97 Tf 6.59 0 Td[(2rEh^)]TJ /F3 11.955 Tf 11.95 0 Td[(=A+=A)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi2r(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B))]TJ /F6 7.97 Tf 6.59 0 Td[(2r22r)]TJ /F6 7.97 Tf 6.58 0 Td[(1Eh^)]TJ /F3 11.955 Tf 11.96 0 Td[(=Ai2r+E(=A)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2r.NowEh^)]TJ /F3 11.955 Tf 11.96 0 Td[(=Ai2r=E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(2rE^)]TJ /F3 11.955 Tf 11.95 0 Td[(2r,E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(2r=[n=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)])]TJ /F6 7.97 Tf 6.59 0 Td[(2r(=n+))]TJ /F6 7.97 Tf 6.59 0 Td[(2r)]TJ /F10 11.955 Tf 8.77 9.69 Td[()]TJ /F8 7.97 Tf 6.67 -4.98 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2)]TJ /F9 11.955 Tf 11.96 0 Td[(2r=22r)]TJ /F10 11.955 Tf 8.77 9.69 Td[()]TJ /F8 7.97 Tf 6.68 -4.98 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2=O(1).WritingQ1fora2m(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)randomvariable,E^)]TJ /F3 11.955 Tf 11.96 0 Td[(2r=2rEQ1 m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+2)]TJ /F9 11.955 Tf 11.96 0 Td[(12r=m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1) m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+22rEQ1 m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F9 11.955 Tf 34.56 8.09 Td[(2 m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)2rm(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1) m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+22r22r)]TJ /F6 7.97 Tf 6.59 0 Td[(1(EQ1 m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1))]TJ /F9 11.955 Tf 11.96 0 Td[(12r+2 m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)2r), 34

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wheretheright-handsideisoftheorderO(m)]TJ /F8 7.97 Tf 6.59 0 Td[(r)sinceE[Q1=[m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)])]TJ /F9 11.955 Tf 11.96 0 Td[(1]2r=O(m)]TJ /F8 7.97 Tf 6.59 0 Td[(r)byLemma 3 .Therefore,Eh^)]TJ /F3 11.955 Tf 11.96 0 Td[(=Ai2r=O(m)]TJ /F8 7.97 Tf 6.59 0 Td[(r).LetQ22m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.ThenE(=A)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2r=B2rEm)]TJ /F9 11.955 Tf 11.95 0 Td[(1 Q2)]TJ /F9 11.955 Tf 11.95 0 Td[(12r=B2rE1)]TJ /F4 11.955 Tf 11.95 0 Td[(Q2=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1) Q2=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)2rB2r"EQ2 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(14r#1 2"EQ2 m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]TJ /F6 7.97 Tf 6.59 0 Td[(4r#1 2=O(1)O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F8 7.97 Tf 6.59 0 Td[(rO(1)=O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F8 7.97 Tf 6.59 0 Td[(r,wherethepenultimatestepfollowsagainfromLemma 3 .Thiscompletestheproof. Lemma 6 impliesthatonecannowobtainresultsparallelto( 3 ),( 3 )andLemma 5 Eh(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^Bi=1+E^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)+E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1=1)]TJ /F9 11.955 Tf 50.88 8.08 Td[(2B m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)+2B2 m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)21+1 n)]TJ /F9 11.955 Tf 11.96 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,wherethelastinequalityfollowssinceE^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=E^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B+o)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1=)]TJ /F9 11.955 Tf 9.3 0 Td[(2B=[m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)]+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,and E^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B2=V^B+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1=2B2 m1+1 n)]TJ /F9 11.955 Tf 11.95 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(3)Theproofofthesecondequationin( 3 )isdeferredtotheAppendix.Inaddition,E(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B2=1+2E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)+3E^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1=1)]TJ /F9 11.955 Tf 50.88 8.09 Td[(4B m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)+6B2 m(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)21+1 n)]TJ /F9 11.955 Tf 11.96 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Hence,V"11)]TJ /F4 11.955 Tf 11.95 0 Td[(B 1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B#=2B221 m(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)21+1 n)]TJ /F9 11.955 Tf 11.95 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1, 35

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"E 1)]TJ /F4 11.955 Tf 11.95 0 Td[(B 1)]TJ /F9 11.955 Tf 13.41 5.32 Td[(^^B)]TJ /F9 11.955 Tf 11.95 0 Td[(1!#2=o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Now,applyingLemma 2 tog2(x)=1=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x)2,x=^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B),a=0andn=0,onegets1=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B2=1=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2+h^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)3iZ10g(1)2h^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)id.ThisimpliesE(1="1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^B2Xi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2#)=E(1="(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2Xi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2#)+E("^^B)]TJ /F4 11.955 Tf 11.95 0 Td[(B (1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)3Pi(zi)]TJ /F9 11.955 Tf 12.11 0 Td[(z)2#Z10g(1)2h^^B)]TJ /F4 11.955 Tf 11.96 0 Td[(B=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)id).SimilartotheproofofLemma 5 ,onecanprovethattheintegraltermiso)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Hence,E2641 1)]TJ /F9 11.955 Tf 13.42 5.32 Td[(^^B2Pi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2375=1=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(B)2E 1 Pi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2!+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1=B m(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2=n+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thuswehavethefollowingtheorem. Theorem3.2. UnderModel 3 and 3 E^^1)]TJ /F3 11.955 Tf 11.95 0 Td[(12=2e=+211+B1+2 n)]TJ /F9 11.955 Tf 11.95 0 Td[(1B=m(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(3)Noticethatcomparedto( 3 ),anadditionalterm221B2=m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(B)2occursinEquation 3 .Thisisduetothepenaltywepayforestimatingtheunknownmeasurementvariance.Further,theunknownparametersB,2e,and1in( 3 )canbeconsistentlyestimatedby^B=n)]TJ /F6 7.97 Tf 6.58 0 Td[(1^(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=Pi(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2or^^B=min^B,(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1),^2e=Pmi=1Pnj=1(yij)]TJ /F9 11.955 Tf 12.24 0 Td[(yi)2=[m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)],^=Pi,j(zij)]TJ /F9 11.955 Tf 12.11 0 Td[(zi)2=[m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)+2]and^^1(given 36

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Table3-1. Performanceofvariousestimateswhenisknown MeanEmpiricalMSEAnalyticalMSE ~1^^1~1^^1^^1=22.463.010.320.070.07=32.253.030.600.140.13=42.073.020.910.210.21 in( 3 )).Therefore,bypluggingintheseconsistentestimatesfortheunknownparameters,oneobtainsasecond-order-correctestimatoroftheMSEapproximation. 3.4SimulationStudyInthissectionweprovideanumericalstudyconcentratedonthesimpleregressionmodelsconsideredinthepresentchapter,andstudytheperformanceoftheMSEofthenewestimates.Weconsiderm=50,0=1,1=3,=5,2e=2,=9and=2,3or4.Wegeneratedyi,ziforeachibasedon( 3 )and( 3 )whenthevarianceofthemeasurementerrorisknown.Wethengenerated^^Band^^1by( 3 ).Table 3-1 givesthemean,MSEofthe1000valuesofthedirectestimator~1andtheproposedestimator^^1.Further,theanalyticalMSEof^^1isprovidedbasedon( 3 ).Underthecasewhenthemeasurementvarianceisunknown,letn=4.Wethengeneratedyij,zijfor1imand1jn,followedby^^Band^^1basedon( 3 ).Again,wecomputedthemeanandtheMSEover1000iterationsforboththenaiveestimatorandtheproposedestimator^^1.TheresultsaregiveninTable 3-2 .Bothtablesindicatethatthedirectestimateisbiasedandthebiasincreasesasthemeasurementvarianceincreases.Incontrast,ourproposedestimatorisunbiasedforallvaluesof.Further,thenewestimateshowsmuchlessvariationascomparedtothenaiveestimate.Thetablesalsoprovideevidencetojustifytheanalyticalresults. 37

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Table3-2. Performanceofvariousestimateswhenisestimated MeanEmpiricalMSEAnalyticalMSE ~1^^1~1^^1^^1=22.843.000.040.0130.013=32.773.000.070.0220.020=42.692.990.110.0280.027 Table3-3. Performanceoftheproposedestimateswithvarying MeanMeanSquaredError CD~1^^1~1^^1 362.133.030.890.29392.393.030.490.22630.862.294.711.43930.621.975.772.29 Basedonareviewer'scomments,wethenconductedasimulationstudytoinvestigatetheconsistencyoftheproposedestimatorwhenthemeasurementvariancevariesforeachi.WeconsideredModel 3 and 3 ,butassumingbothandtovaryforeachi.First,wegeneratedxiindN(5,i),uiindN(0,i)andeiiidN)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(0,2ewhereiiidIG(C,1),iiidIG(D,1),1imand2e=2.Again,letm=50,0=1and1=3.Wethengeneratedyi,zibasedon( 3 )and( 3 ),^^Bi=min(^Bi,(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1))where^Bi=i(m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)=Pmi=1(zi)]TJ /F9 11.955 Tf 12.1 0 Td[(z)2and^^1=Pmi=1yi)]TJ /F9 11.955 Tf 5.67 -9.68 Td[(^xiJS+)]TJ /F9 11.955 Tf 12.14 1.69 Td[(^xJS+=Pmi=1)]TJ /F9 11.955 Tf 5.67 -9.68 Td[(^xiJS+)]TJ /F9 11.955 Tf 12.14 1.69 Td[(^xJS+2where^xiJS+=1)]TJ /F9 11.955 Tf 13.41 5.31 Td[(^^Bizi+^^Biz,1im.WecalculatedthemeanandtheMSEover1000iterationsforseveralgroupsofdistinctvaluesoffC,Dgforboththedirectestimator~1andthenewestimator^^1.Table 3-3 reportstheresults. 38

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Itisinterestingtonotethat^^1appearstobeunbiasedfor1andhassmallMSEwhenC
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CHAPTER4LIKELIHOODBASEDMETHODSFORSMALLAREAESTIMATIONWITHCOVARIATESSUBJECTTOMEASUREMENTERROR 4.1BackgroundSmallareaestimationisatopicofgrowingimportanceinsurveysampling.Theneedforsuchestimatesisfeltbothinthepublicandprivatesectors.Thedirectsmallareasurveyestimatesareusuallyveryunstable,beingaccompaniedwithlargestandarderrorsandcoefcientsofvariation.Thismakesitanecessitytoborrowstrength,orconnectdifferentsmallareasthroughsomemodels.Thesemodelsprovidethenecessarylinkbybringinginrelevantauxiliaryinformation,oftencollectedfrommultipleadministrativesources.Often,however,theauxiliaryrandomvariablesaremeasuredwitherror.Tociteanexample,asgiveninFuller(1987,p.2),suppose,wewanttopredicttheyieldofcorninseveralcountiesinIowaandthecovariateusedisavailablenitrogeninthesoil.Toestimatetheavailablesoilnitrogen,itisnecessarytosamplethesoiloftheexperimentalplot,andtoperformalaboratoryanalysisontheselectedsample.Asaresultofthesamplingandofthelaboratoryanalysis,wedonotobservethetrueavailablenitrogen,butonlyitsestimate.Literatureonsmallareaestimationwithcovariatessubjecttomeasurementerrorissomewhatsparse.GhoshandSinha(2006,2007)addressedthisproblemunderaunitlevelstructural(whentheunobservedcovariatesarestochastic)andfunctional(whenthecovariatesarenonstochastic)measurementerrormodels.Torabi,DattaandRao(2009)improvedtheresultsofGhoshandSinha(2006)forthestructuralmeasurementerrormodels.Incontrast,YbarraandLohr(2008)consideredanarealevelmodelinthesmallareacontextwithauxiliaryinformationsubjecttomeasurementerror.TheyadjustedveryappropriatelythemethodofmomentapproachofPrasadandRao(1990)toestimatethemodelparameters.However,themethodofmomentsapproachhasoftenbeencriticizedonthegroundofitslackofefciencyascomparedtothemaximum 40

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likelihood(ML)orrestrictedmaximumlikelihood(REML)approach(seeforexampleDattaandLahiri,2000)forparameterestimation.Weconsiderinthischapterabiascorrectedprolelikelihoodapproachforobtainingsmallareaestimateswhencovariatesaresubjecttomeasurementerror.ThegeneralapproachisdescribedinSection 4.2 ofthischapter.Section 4.3 considersmeansquarederror(MSE)ofourestimators.Firstorderasymptoticoptimalityoftheseestimatorsisproved,andestimatorsoftheseMSE'sareobtained.Section 4.4 providesabriefsimulationstudywhichsubstantiatessomeofthetheoreticalndingsoftheprevioussection.SomenalremarksaremadeinSection 4.5 4.2TheprolelikelihoodapproachConsiderthefollowinggeneralmixedlinearmodelwheretheauxiliaryinformationxiismeasuredwitherror yi=xTi+ui+ei,Xi=xi+i,i=1,,m.(4)Therandomeffectsui,thesamplingerroreiandthemeasurementerroriaremutuallyindependentwitheiindN(0,Di),uiiidN)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(0,2uandiindN(0,Ci).Further,isanite-p-dimensionalvectorofunknownparameters,2uisunknownandneedtobeestimated.TheparametersCiandDi,i=1,,mareassumedtobeknown.WealsoallowsomeorallelementsofCitobezero.Thuscaseswithsomeornomeasurementerrorincovariatesiscovered.TherstpartofModel 4 isusuallyreferredtoastheFay-Herriotmodel(FayandHerriot,1979).Thesemodelsarealsoreferredtoasarea-levelmodels,andarequiteusefulintheabsenceofunitleveldata.Thexi'sarenon-stochasticbutunknown.ThusModel 4 isafunctionalmeasurementerrormodelincontrasttoastructuralmeasurementerrormodelwherethexi'sarestochastic(Fuller,1987).Weareinterestedinpredictingthegeneralmixedeffectsi=xTi+ui,whereinsmallareacontext,icouldbethecountofpoorschool-agechildrenintheitharea 41

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(smallareatotals)orpercapitaincomeforsmallplacesintheUnitedStates(smallareameans).Itiswell-knownthatintheabsenceofmeasurementerrorandknown,thebestlinearunbiasedpredictor(BLUP)ofiisgivenby ~~i=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Ai)yi+AixTi,Ai=Di=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u,i=1,,m.(4)ThisisalsotheBayesestimatoroftheiwhenonerewritestherstpartofModel 4 asyijiindN(i,Di)andiindN)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(xTi,2u,i=1,,m.Sincethexiareunknown,thenumberofnuisanceparametersinModel 4 ismp.Toreducethenumberofnuisanceparameters,weadopttheprolelikelihoodapproach,rstobtainingthelikelihoodafterprolingthexi.Tothisend,webeginwiththemodel 0B@yiXi1CAiidN2640B@xTixi1CA,0B@Di+2u00Ci1CA375,i=1,...,m.(4)Thelog-likelihoodfunctionisthengivenbyl)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(x1,,xm,,2u=)]TJ /F9 11.955 Tf 11.29 -.16 Td[([(m+pm)=2]log2)]TJ /F9 11.955 Tf 13.15 8.09 Td[(1 2mXi=1logjCij)]TJ /F9 11.955 Tf 19.12 8.09 Td[(1 2mXi=1log)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F9 11.955 Tf 10.49 8.08 Td[(1 2mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(xTi2=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F9 11.955 Tf 13.15 8.08 Td[(1 2mXi=1(Xi)]TJ /F4 11.955 Tf 11.96 0 Td[(xi)TC)]TJ /F6 7.97 Tf 6.59 0 Td[(1i(Xi)]TJ /F4 11.955 Tf 11.95 0 Td[(xi).Werewrite)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(xTi2=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u+(Xi)]TJ /F4 11.955 Tf 11.96 0 Td[(xi)TC)]TJ /F6 7.97 Tf 6.59 0 Td[(1i(Xi)]TJ /F4 11.955 Tf 11.95 0 Td[(xi)=xTih)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F6 7.97 Tf 6.58 0 Td[(1T+C)]TJ /F6 7.97 Tf 6.58 0 Td[(1iixi)]TJ /F9 11.955 Tf 9.3 0 Td[(2h)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F6 7.97 Tf 6.58 0 Td[(1yi+C)]TJ /F6 7.97 Tf 6.59 0 Td[(1iXiiTxi+y2i=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u+XTiC)]TJ /F6 7.97 Tf 6.58 0 Td[(1iXi. 42

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First,holdand2uxed,andwrite!i=Di+2u+TCi,applyingthestandardmatrixinversionformula(Rao,1973,p.33),theMLEofxiis ^xi=h)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F6 7.97 Tf 6.59 0 Td[(1T+C)]TJ /F6 7.97 Tf 6.59 0 Td[(1ii)]TJ /F6 7.97 Tf 6.58 0 Td[(1h)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F6 7.97 Tf 6.59 0 Td[(1yi+C)]TJ /F6 7.97 Tf 6.58 0 Td[(1iXii="Ci)]TJ /F4 11.955 Tf 16.4 8.85 Td[(Ci)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F6 7.97 Tf 6.58 0 Td[(1TCi 1+(Di+2u))]TJ /F6 7.97 Tf 6.58 0 Td[(1TCi#hC)]TJ /F6 7.97 Tf 6.59 0 Td[(1iXi+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F6 7.97 Tf 6.59 0 Td[(1yii=Xi)]TJ /F4 11.955 Tf 13.15 8.09 Td[(CiTXi !i+Ciyi Di+2u)]TJ /F3 11.955 Tf 26.91 8.09 Td[(TCi !i(Di+2u)Ciyi=Xi)]TJ /F4 11.955 Tf 13.15 8.09 Td[(CiTXi !i+Ciyi !i=Xi+Ci !i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi. (4) Substitutionof^xiforxiin( 4 )resultsinthepredictor~ifori,andisgivenby~i~i)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(,2u=1)]TJ /F4 11.955 Tf 26.46 8.08 Td[(Di Di+2uyi+Di Di+2uXTi+TCi !i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi=1)]TJ /F4 11.955 Tf 26.46 8.09 Td[(Di Di+2u1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(TCi !iyi+Di Di+2u1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(TCi !iXTi=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi)yi+BiXTi,whereBi=Di=!i,i=1,...,m.Notethat~icoincideswiththeintermediateestimatorasconsideredinYbarraandLohr(2008).Intheirpaper,theyareinterestedtondtheestimatorwithaminimumMSEamongalllinearcombinationsoftheform(1)]TJ /F4 11.955 Tf 11.96 0 Td[(ai)yi+aiXTi,0ai1.However,~icannot,ingeneral,beusedasapredictorofisincebothand2uaretypicallyunknown.Inordertoestimatetheseparameters,weusetheprolelikelihoodapproach,prolingthenuisanceparametersxibytheirMLE's^xi.Writingmasagenericconstantnotdependingonand2u,theprolelikelihoodlp)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(,2ul)]TJ /F9 11.955 Tf 5.66 -9.68 Td[(^x1,,^xm,,2uisgivenbylp)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(,2u=m)]TJ /F9 11.955 Tf 13.15 8.09 Td[(1 2mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xTi2=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F9 11.955 Tf 13.15 8.08 Td[(1 2mXi=1(Xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)TC)]TJ /F6 7.97 Tf 6.59 0 Td[(1i(Xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi). 43

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Theexpression)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xTi2)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F6 7.97 Tf 6.59 0 Td[(1+(Xi)]TJ /F9 11.955 Tf 12.13 0 Td[(^xi)TC)]TJ /F6 7.97 Tf 6.59 0 Td[(1i(Xi)]TJ /F9 11.955 Tf 12.14 0 Td[(^xi)simpliesto )]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(TCi !i2+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2TCi !2i=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F4 11.955 Tf 12.96 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2=!2i+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2TCi=!2i=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2=!i. (4) Hence,by( 4 )and( 4 ), lp)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(,2u=m)]TJ /F9 11.955 Tf 13.15 8.09 Td[(1 2mXi=1log)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Di+2u)]TJ /F9 11.955 Tf 13.15 8.09 Td[(1 2mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2 !i.(4)Theresultingscorefunctionsforand2ubasedon( 4 )arethusgivenbym)]TJ /F6 7.97 Tf 6.58 0 Td[(1@lp @=m)]TJ /F6 7.97 Tf 6.58 0 Td[(1"mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiXi !i+mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2Ci !2i#andm)]TJ /F6 7.97 Tf 6.59 0 Td[(1@lp @2u=)]TJ /F9 11.955 Tf 15.48 8.09 Td[(1 2mmXi=11 Di+2u+1 2mmXi=1)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2 !2i.WemaynotethatE(@lp=@)=0butE)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(@lp=@2u=)]TJ /F6 7.97 Tf 14.01 4.71 Td[(1 2mPmi=1h)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(!)]TJ /F6 7.97 Tf 6.59 0 Td[(1ii6=0.Itiswell-knownthatbiasedestimatingfunctionsofthistypewillleadtoinconsistentMLE's.Thisisnotsurprisingsinceinthiscasethenumberofnuisanceparametersisgrowingindirectproportiontothesamplesize-anotherinstanceoftheNeyman-Scottphenomenon.Toavoidthis,weconsiderthebias-correctedprolescorefunction Sm( )=0B@S1m( )S2m( )1CA=m)]TJ /F6 7.97 Tf 6.59 0 Td[(10B@Pmi=1(yi)]TJ /F8 7.97 Tf 6.58 0 Td[(XTi)Xi !i+Pmi=1(yi)]TJ /F8 7.97 Tf 6.59 0 Td[(XTi)2Ci !2i1 2Pmi=1(yi)]TJ /F8 7.97 Tf 6.59 0 Td[(XTi)2 !2i)]TJ /F6 7.97 Tf 14.82 4.71 Td[(1 !i1CA.(4)Writing^ =^T,^2uTasthesolutionofSm( )=0,wenowobtainthenalpredictorofias^i~i^,^2u=1)]TJ /F9 11.955 Tf 13.41 2.66 Td[(^Biyi+^BiXTi^,where^Bi=Di=Di+^2u+^TCi^,i=1,,m.Inordertostudythemeansquarederrorandotherpropertiesof^i,weneedanumberofpreliminarycalculationswhichwestateintheformoflemmas. 44

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Lemma7. yi)]TJ /F4 11.955 Tf 9.79 0 Td[(XTiandXi+(yi)]TJ /F8 7.97 Tf 6.59 0 Td[(XTi)Ci !iareindependentlydistributedforalli=1,,m. Proof. Cov yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi,Xi+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiCi !i!=Cov)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi,Xi+!)]TJ /F6 7.97 Tf 6.58 0 Td[(1iV)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiTCi=)]TJ /F3 11.955 Tf 9.3 0 Td[(TCi+TCi=0.Theindependenceresultfollowsnowfromthenormalityandlinearityofthefunctions. Lemma8. ESm( )STm( )=264m)]TJ /F6 7.97 Tf 6.59 0 Td[(2Pmi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(1iCi)]TJ /F8 7.97 Tf 13.15 5.25 Td[(CiTCi !i+xixTi001 2m)]TJ /F6 7.97 Tf 6.58 0 Td[(2Pmi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2i375. Proof. ESm( )STm( )=E264S1m( )ST1m( )S1m( )S2m( )S2m( )ST1m( )S22m( )375. (4) First, ES1m( )ST1m( )=m)]TJ /F6 7.97 Tf 6.59 0 Td[(2mXi=1V")]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiXi !i+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2Ci !2i#=m)]TJ /F6 7.97 Tf 6.59 0 Td[(2mXi=1V")]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi !i(Xi+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiCi !i)#. (4) ByLemma 7 andtheformulaV(Z1Z2)=V(Z1)V(Z2)+V(Z1)E(Z2)ET(Z2)+V(Z2)E2(Z1)forascalarZ1independentlydistributedofavectorZ2,therighthandsideof( 4 )simpliestoV")]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi !i#V"Xi+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiCi !i#+V")]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi !i#E Xi+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiCi !i!"E Xi+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiCi !i!#T+V"Xi+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiCi !i#"E()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi !i)#2. (4) 45

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ButE)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2=!iandV"Xi+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiCi !i#=V24Xi)]TJ /F4 11.955 Tf 11.95 0 Td[(xi+n)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(xTi)]TJ /F9 11.955 Tf 11.96 -.17 Td[((Xi)]TJ /F4 11.955 Tf 11.95 0 Td[(xi)ToCi !i35=Ci+CiTCi !i)]TJ /F9 11.955 Tf 13.15 8.08 Td[(2CiTCi !i=Ci)]TJ /F4 11.955 Tf 13.15 8.08 Td[(CiTCi !i.Further,E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi=0andEXi+(yi)]TJ /F8 7.97 Tf 6.59 0 Td[(XTi)Ci !i=xi.Hence,from( 4 )and( 4 ), ES1m( )ST1m( )=m)]TJ /F6 7.97 Tf 6.58 0 Td[(2mXi=1V)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(!)]TJ /F6 7.97 Tf 6.58 0 Td[(1iCi)]TJ /F3 11.955 Tf 11.95 0 Td[(!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iCiTCi+!)]TJ /F6 7.97 Tf 6.59 0 Td[(1ixixTi (4) and ES22m( )=1 4m)]TJ /F6 7.97 Tf 6.58 0 Td[(2mXi=1Vh!)]TJ /F6 7.97 Tf 6.59 0 Td[(2i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2i=1 2m)]TJ /F6 7.97 Tf 6.59 0 Td[(2mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2i. (4) Finally,E[S1m( )S2m( )]=1 2m)]TJ /F6 7.97 Tf 6.58 0 Td[(2mXi=1E")]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiXi !i+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2Ci !2i#")]TJ /F4 11.955 Tf 5.47 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2 !2i)]TJ /F9 11.955 Tf 15.24 8.09 Td[(1 !i#=1 2m)]TJ /F6 7.97 Tf 6.58 0 Td[(2mXi=1E")]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi3Xi !3i)]TJ /F10 11.955 Tf 13.15 18.53 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiXi !2i+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi4Ci !4i)]TJ /F10 11.955 Tf 13.15 18.53 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2Ci !3i#=1 2m)]TJ /F6 7.97 Tf 6.58 0 Td[(2mXi=1)]TJ /F9 11.955 Tf 10.49 8.09 Td[(3Ci !2i+Ci !2i+2Ci !2i=0,since Eh)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi3Xii=E"(Xi+)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiCi !i)]TJ /F10 11.955 Tf 13.15 18.53 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiCi !i))]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi3#=)]TJ /F4 11.955 Tf 9.3 0 Td[(E")]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi4Ci !i#=)]TJ /F9 11.955 Tf 9.3 0 Td[(3Ci!i. (4) Theresultfollowsnowfrom( 4 )and( 4 )-( 4 ).Thenextlemmaisusefulinndingthesandwichinformationmatrix. 46

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Lemma9. Hm( )def=264E)]TJ /F12 7.97 Tf 10.49 5.59 Td[(@S1m( ) @TE)]TJ /F12 7.97 Tf 10.49 5.59 Td[(@S1m( ) @2uE)]TJ /F12 7.97 Tf 10.49 5.59 Td[(@S2m( ) @TE)]TJ /F12 7.97 Tf 10.49 5.59 Td[(@S2m( ) @2u375=264m)]TJ /F6 7.97 Tf 6.59 0 Td[(2Pmi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(1ixixTim)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iCi01 2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2i375. Proof. @S1m( ) @T=)]TJ /F4 11.955 Tf 9.3 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(1iXiXTi)]TJ /F9 11.955 Tf 11.95 0 Td[(2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1Xi)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi !2iTCi)]TJ /F9 11.955 Tf 9.3 0 Td[(2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiCiXTi !2i+m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2Ci !2i)]TJ /F9 11.955 Tf 9.3 0 Td[(4m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2CiTCi !3i.Hence, E)]TJ /F3 11.955 Tf 10.5 8.09 Td[(@S1m( ) @T=m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(1i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(Ci+xixTi)]TJ /F9 11.955 Tf 11.95 0 Td[(2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iCiTCi)]TJ /F9 11.955 Tf 9.29 0 Td[(2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2iCiTCi)]TJ /F4 11.955 Tf 11.95 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(1iCi+4m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2iCiTCi=m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(1ixixTi. (4) Further, E)]TJ /F3 11.955 Tf 10.49 8.09 Td[(@S1m( ) @2u=m)]TJ /F6 7.97 Tf 6.58 0 Td[(1E"mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTiXi !2i+2mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2Ci !3i#=)]TJ /F4 11.955 Tf 9.3 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2iCi+2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2iCi=m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2iCi, (4) 47

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E)]TJ /F3 11.955 Tf 10.49 8.08 Td[(@S2m( ) @T=m)]TJ /F6 7.97 Tf 6.59 0 Td[(1E"mXi=1)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTiXTi !2i+2mXi=1)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2TCi !3i)]TJ /F8 7.97 Tf 17.08 14.94 Td[(mXi=1TCi !2i#=)]TJ /F4 11.955 Tf 9.3 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iTCi+2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iTCi)]TJ /F4 11.955 Tf 11.95 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iTCi=0 (4) and E)]TJ /F3 11.955 Tf 10.5 8.08 Td[(@S2m( ) @2u=m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1E")]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2 !3i)]TJ /F9 11.955 Tf 19.48 8.08 Td[(1 2!2i#=1 2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2i.(4)Combine( 4 )-( 4 )togetLemma 9 Beforeproceedingfurther,weneedafewassumptionstocontroltheasymptoticbehavioroftheelementsofmES1m( )ST1m( )andHm( ).Inparticular,weassume(i)m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(1ixixTi!A,(ii)m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1!)]TJ /F6 7.97 Tf 6.58 0 Td[(2i!g,(iii)m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(!)]TJ /F6 7.97 Tf 6.59 0 Td[(1iCi)]TJ /F3 11.955 Tf 11.96 0 Td[(!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iCiTCi!Fand(iv)m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1!)]TJ /F6 7.97 Tf 6.59 0 Td[(2iCi!dasm!1.Withtheseassumptionsasm!1 mES1m( )ST1m( )!264A+FddT1 2g375=V1(say)(4)and Hm( )!264Ad01 2g375=V2(say).(4)Thenexttheoremprovidestheasymptoticdistributionofp m^ )]TJ /F3 11.955 Tf 11.95 0 Td[( Theorem4.1. Assume(i))]TJ /F9 11.955 Tf 11.95 0 Td[((iv).Thenp m^ )]TJ /F3 11.955 Tf 11.96 0 Td[( d!N(0,V),whereV=0B@A)]TJ /F6 7.97 Tf 6.59 0 Td[(1(A+F)A)]TJ /F6 7.97 Tf 6.59 0 Td[(1+2g)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.58 0 Td[(1ddTA)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F9 11.955 Tf 9.3 0 Td[(2g)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(1d)]TJ /F9 11.955 Tf 9.3 0 Td[(2g)]TJ /F6 7.97 Tf 6.58 0 Td[(1dTA)]TJ /F6 7.97 Tf 6.59 0 Td[(12g)]TJ /F6 7.97 Tf 6.59 0 Td[(11CA. 48

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Proof. BytherstorderTaylorexpansion,0=Sm^ =Sm( )+@Sm( ) @ Tj =~ ^ )]TJ /F3 11.955 Tf 11.96 0 Td[( ,where~ )]TJ /F3 11.955 Tf 11.95 0 Td[( <^ )]TJ /F3 11.955 Tf 11.95 0 Td[( .Inviewof(i))]TJ /F9 11.955 Tf 12.16 0 Td[((iv),applyingaresultofFoutz(1976),^ p! asm!1.Hence,bythecontinuityofthefunction@Sm( ) @ T,andassumptions(i))]TJ /F9 11.955 Tf 12.28 0 Td[((iv),@Sm( ) @ Tj =~ )]TJ /F12 7.97 Tf 10.5 5.59 Td[(@Sm( ) @ Tp!0while)]TJ /F10 11.955 Tf 11.29 13.27 Td[(@Sm( ) @ T)]TJ /F6 7.97 Tf 6.59 0 Td[(1p!V)]TJ /F6 7.97 Tf 6.59 0 Td[(12asm!1by( 4 ).Now,p m^ )]TJ /F3 11.955 Tf 11.96 0 Td[( =)]TJ /F3 11.955 Tf 10.49 8.09 Td[(@Sm( ) @ Tj =~ )]TJ /F6 7.97 Tf 6.59 0 Td[(1p mSm( ).Againdueto(i))]TJ /F9 11.955 Tf 12.97 0 Td[((iv)andthemultiparametercentrallimittheoremp mSm( )d!N(0,V1)by( 4 ).Thisleadstop m^ )]TJ /F3 11.955 Tf 11.95 0 Td[( d!N)]TJ /F9 11.955 Tf 5.48 -9.68 Td[(0,V)]TJ /F6 7.97 Tf 6.59 0 Td[(12V1V)]TJ /F6 7.97 Tf 6.59 0 Td[(12andonsimplicationV)]TJ /F6 7.97 Tf 6.59 0 Td[(12V1V)]TJ /F6 7.97 Tf 6.58 0 Td[(12=V. Remark5.NotethatifCi=0fori=1,,m,thatis,whenthereisnomeasurementerrorinthemodel.Then,Vreducesto0B@A)]TJ /F6 7.97 Tf 6.59 0 Td[(1002g)]TJ /F6 7.97 Tf 6.59 0 Td[(11CA,whichmeansand2uareorthogonalinthesenseofCoxandReid(1987)andHuzurbazar(1950).ThisagreeswiththeresultofDattaandLahiri(2000).Remark6.Besidestheprolelikelihoodapproachappliedinhere,onecanalsoutilizetheintegratedlikelihoodapproach(REMLapproachisaspecialcase)toeliminatethenuisanceparametersinthemodel.Alternatively,integratingoutthexi'sthelogintegratedlikelihoodfor isthesameaslp( )exceptlog)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2uisreplacedbylog(!i).Fromthisonecansimilarlyobtainscorefunctionsforand2ugivenby@lI @=)]TJ /F10 11.955 Tf 11.29 8.97 Td[(Pmi=1Ci !i+Pmi=1(yi)]TJ /F8 7.97 Tf 6.58 0 Td[(XTi)Xi !i+Pmi=1(yi)]TJ /F8 7.97 Tf 6.59 0 Td[(XTi)2Ci !2iand@lI @2u=)]TJ /F6 7.97 Tf 10.49 4.71 Td[(1 2Pmi=11 !i+1 2Pmi=1(yi)]TJ /F8 7.97 Tf 6.59 0 Td[(XTi)2 !2i.Incontrasttotheprolelikelihoodapproach,thescorefunctionisnowunbiasedfor2u,whilebiasedfor.Notethatifweonceagaincorrectforthebiasinthescorefunction,wewillobtaintheadjustedscorefunctionidenticaltotheonegivenin( 4 ).Thereforetheestimatorobtainedfrom( 4 )canalsobederivedfromtheintegratedlikelihoodapproach. 49

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4.3ThemeansquarederrorInthissection,wewillprovetherstorderasymptoticoptimalityfortheestimator^iinthesenseofRobbins(1956)under( 4 ).Westartwiththeequality,MSE(^i)=E(~i)]TJ /F3 11.955 Tf 11.96 0 Td[(i+^i)]TJ /F9 11.955 Tf 12.68 2.66 Td[(~i)2=E(~i)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2+E(^i)]TJ /F9 11.955 Tf 12.69 2.66 Td[(~i)2+2E(~i)]TJ /F3 11.955 Tf 11.95 0 Td[(i)(^i)]TJ /F9 11.955 Tf 12.69 2.66 Td[(~i). (4)ButE(~i)]TJ /F3 11.955 Tf 11.96 0 Td[(i)2=E((1)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi)yi+BiXTi)]TJ /F3 11.955 Tf 11.95 0 Td[(i)2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi)2Di+B2i(2u+TCi)=Di(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi). (4)Itisnotdifculttonotethatasymptotically,thisrstcomponentistheonlyO(1)termin( 4 ).Infact,wewillshowinthenexttheoremthatundercertainconditionsm)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1E(~i)]TJ /F9 11.955 Tf 12.68 2.66 Td[(^i)2!0asm!1.Thiswillimplyjm)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1E(~i)]TJ /F3 11.955 Tf 9.3 0 Td[(i)(^i)]TJ /F9 11.955 Tf 10.03 2.66 Td[(~i)j"m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1E1 2(~i)]TJ /F3 11.955 Tf 11.95 0 Td[(i)2#"m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1E1 2(^i)]TJ /F9 11.955 Tf 12.68 2.66 Td[(~i))2#!0asm!1whenoneusesinaddition( 4 )assuming!i=O(1),i=1,,m. Theorem4.2. Assume(i)!i,!)]TJ /F6 7.97 Tf 6.59 0 Td[(1i,xiandtheelementsinCi,i=1,,mareuniformlybounded,(ii)supm1E^)]TJ /F3 11.955 Tf 11.95 0 Td[(4+<1forsome>0.Then,m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1E(^i)]TJ /F9 11.955 Tf 12.68 2.65 Td[(~i)2!0asm!1. Proof. Writing^i)]TJ /F9 11.955 Tf 12.69 2.66 Td[(~i=1)]TJ /F9 11.955 Tf 13.41 2.66 Td[(^Biyi+^BiXTi^)]TJ /F10 11.955 Tf 11.95 13.27 Td[(h(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi)yi+^BiXTii=)]TJ /F10 11.955 Tf 11.29 13.28 Td[(^Bi)]TJ /F4 11.955 Tf 11.96 0 Td[(Bih)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi^)]TJ /F3 11.955 Tf 11.96 0 Td[(i+BiXTi^)]TJ /F3 11.955 Tf 11.95 0 Td[(. 50

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Usingtheelementaryinequality(a+b+c)23)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(a2+b2+c2,wehave m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1E(^i)]TJ /F9 11.955 Tf 12.69 2.65 Td[(~i)23m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1E[^Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2+^Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2XTi^)]TJ /F3 11.955 Tf 11.96 0 Td[(2+B2iXTi^)]TJ /F3 11.955 Tf 11.95 0 Td[(2]. (4) Now^Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi=Di=Di+^2u+^TCi^)]TJ /F4 11.955 Tf 11.96 0 Td[(Di=)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(Di+2u+TCi=Di2u)]TJ /F9 11.955 Tf 14.82 3.03 Td[(^2u+TCi)]TJ /F9 11.955 Tf 13.43 2.65 Td[(^TCi^=!i^!i!)]TJ /F6 7.97 Tf 6.59 0 Td[(1ih^2u)]TJ /F3 11.955 Tf 11.95 0 Td[(2u+^TCi^)]TJ /F3 11.955 Tf 11.96 0 Td[(TCiiP!0since^P!,^2uP!2uandmax1im!)]TJ /F6 7.97 Tf 6.59 0 Td[(1i)]TJ /F6 7.97 Tf 6.58 0 Td[(2u.Further,m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2=m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1!i=O(1).Hence,m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1^Bi)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi2)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2P!0.Nowm)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1^Bi)]TJ /F4 11.955 Tf 11.96 0 Td[(Bi2)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2max1im^Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi2=:Jmwhichisuniformlyintegrablesincesupm1E)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(J2msupm1m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1E)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(yi)]TJ /F4 11.955 Tf 11.95 0 Td[(XTi4=3supm1m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1!2i3maxi1!2i<1.Hence,E"m)]TJ /F6 7.97 Tf 6.58 0 Td[(1mXi=1^Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(yi)]TJ /F4 11.955 Tf 11.96 0 Td[(XTi2#!0.NextweshowthatEm)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1^Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2hXTi^)]TJ /F3 11.955 Tf 11.95 0 Td[(i2!0.ApplyingtheCauchy-Schwarzinequality,wehave 51

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E24(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1~Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2hXTi^)]TJ /F3 11.955 Tf 11.95 0 Td[(i2)1+ 435E24(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1kXik2^)]TJ /F3 11.955 Tf 11.96 0 Td[(2)1+ 435E1 224(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1mXi=1kXik2)2+ 235E1 2^)]TJ /F3 11.955 Tf 11.96 0 Td[(4+.Sincemax1imEnm)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pmi=1kXik2o2+ 2isboundedabove,wehaveproveduniformintegrabilityofm)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1~Bi)]TJ /F4 11.955 Tf 11.95 0 Td[(Bi2hXTi^)]TJ /F3 11.955 Tf 11.95 0 Td[(i2,m1.Asimilarargumentgivestheuniformintegrabilityofthelasttermin( 4 )inm,thusprovingTheorem 4.2 Theorem 4.2 indicatesthatm)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1MSE(^i))]TJ /F4 11.955 Tf 12.76 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pmi=1[Di(1)]TJ /F4 11.955 Tf 12.77 0 Td[(Bi)]!0asm!1,henceprovingtherstorderoptimaloftheproposedpredictorinthesenseofRobbins(1956).Weplugintheestimator^ of directlyinto( 4 )togetanestimatoroftheMSEgivenby mse(^i)=Di(1)]TJ /F9 11.955 Tf 13.42 2.65 Td[(^Bi)=Dih1)]TJ /F4 11.955 Tf 11.95 0 Td[(Di=Di+^2u+^TCi^i.(4)Itiseasytocheckthatmse(^i))]TJ /F4 11.955 Tf 12.62 0 Td[(MSE(^i)P!0asm!1.NotethatthisrstorderestimatoroftheMSEisgreaterthanitscounterpartwithnomeasurementerrorinthecovariates:Di2u=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Di+2u,duetotheadditionalvariationintroducedbythepresenceofthemeasurementerror. 4.4SimulationWeconductedanumericalstudytoevaluatetheperformanceofthenewestimate.Thesimplebalancedcasewasconsidered.WerstgeneratedzifromaN(5,9)distribution,uifromaN(0,4)distributionandeifromaN(0,9)distribution.Letp=2,m=50andxTi=(1,zi).Also,assumingCi=0B@000c1CA,wherec=1,3or6.Thus, 52

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Table4-1. PerformanceoftheMSEofvariousestimates EmpiricalMeanSquaredErrorAnalyticalMeanSquaredError yi~~i(~ )~~i(^ )^i^ic=19.043.347.896.066.13c=38.893.3215.007.107.12c=69.003.3721.627.957.83 includingcaseswithnomeasurementerrorinoneofthecovariates.Conditionalonzi,wethengeneratedi=1+3zi+ui,yi=i+eiandXi=xi+i.Wecompared(i)thedirectestimatoryi,(ii)theEBLUP~~i(~ )discussedwheretheplugged-inestimator~ istheMLestimateof whenthetruecovariatesxi'sinthemodelareknown(DattaandLahiri,2000),(iii)theEBLUP~~i(^ )where^ istheMLestimateof whenusingtheobservedcovariatesXi,and(iv)thenewestimate^ibasedonthousanditerations.Wealsoprovidedtheanalyticalresultsfortherst-orderestimatorofthemeansquarederrorgivenin( 4 ).Table 4-1 showsthattheproposedestimatorperformedbetterthanboththedirectestimatorandtheEBLUP~~i(^ )undervariedmeasurementerror.TheMSEfortheproposedestimatorgetsclosertothedirectestimatorasthemeasurementvariancecincreases.Further,theEBLUP~~i(^ )whichignoresthemeasurementerrorperformstheworstforlargecamongalltheestimatorsconsidered.TheresultvalidatesthefactthattheMSEgivenin( 4 )isgreaterthanitscounterpartwithnomeasurementerrorinthecovariates,whileillustratesafairlygoodanalyticalapproximationtotheempiricalMSE. 4.5SummaryInthischapter,weemployedtheprolelikelihoodapproachindealingwiththemeasurementerrorprobleminsmallareacontext.WeconsideredadjustedMLandREMLmethodsinplaceofthePrasadandRao(1990)methodofmomentsapproach 53

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toobtainamorestableestimator.Wealsoexploredrstorderasymptoticallyoptimalestimationoftheproposedpredictorinthischapter. 54

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CHAPTER5SUMMARYANDCONCLUSIONSThisDissertationconcernsthemeasurementerrorestimationandpredictionproblemsthatareusefulinmanyareasofapplication.Therstpartconsidersanestederrorlinearregressionmodelwithmultiplecovariatessubjecttomeasurementerror.Acorrectsecond-orderunbiasedapproximationtotheMSEoftheEBpredictorisobtained.However,theresultsarelimitedtothebalancedcase.Toallowforabroaderapplicationinpractice,forinstanceinsmallareaproblems,afuturetopicofresearchmightbetoextendtheresultstotheunbalancedcase.ThesecondpartinvolvesaninterestingyeteasyapproachofmodifyingtheinconsistencyoftheusualleastsquaresestimatorinmeasurementerrorregressionproblemsusinganalternativeJames-Steinestimator.WeprovidearigoroussecondorderexpansionofthemeansquarederroroftheproposedJames-Steinestimatorbothunderknownmeasurementvarianceandunknownmeasurementvariance.Onemightextendtheresulttothecasewhentherearemultiplecovariatessubjecttomeasurementerror,withthehelpoftheTaylorseriesexpansionofascalar-valuedfunctionofvectorvariables.Anonlinearoranon-Gaussianmodelmightalsobeaninterestingtopicofconcern.ThenalpartofthisDissertationintroducesanewlikelihood-basedapproachindealingwiththemeasurementerrorproblem.AnadjustedMLestimationisemployedusingthemodiedscorefunctionsderivedfromaprolelikelihood.WeprovedtherstorderasymptoticallyoptimaloftheMSEofthepredictedmixedeffectsinthesenseofRobbins(1956).Datta,RaoandSmith(2005)showedfortheFay-Herriotmodelwithoutmeasurementerrorthatthemaximumlikelihood(ML)andtheresidualmaximumlikelihood(REML)estimatorsof2ustudiedbyDattaandLahiri(2000),andanestimatingequationbasedestimatorof2usuggestedindependentlybyFayand 55

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Herriot(1979)andPrefermentandNatant(1981)(FD-PAN)aremorestablethanthePrasad-Raoestimator.YettheFD-PANestimatorshowsbetterstabilityandrobustnessamongalltheseestimators.Therefore,analternativeapproachtoestimatethemodelparametersbesidestheproleandintegratedlikelihoodmethodsconsideredherewouldbeanextensionoftheFD-PANestimatingequationfor2u.ThisapproachisbasedonweightedleastsquaresforbyminimizingQ)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(,2uforxed2uandbysolvingamodiedFD-PANestimatingequationfor2ugivenbyQ)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(,2u=m.Theseestimatingequationsareallunbiased.TheFD-PANestimatingequationislikelytobeverysimilartothescorefunctionof2ubasedontheintegratedlikelihood. 56

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APPENDIXADERIVATIONOFTHEOREM2.3First,writing^(S)=SSB)]TJ /F6 7.97 Tf 6.58 0 Td[(1XSSByX,^PBi)]TJ /F9 11.955 Tf 12.68 2.65 Td[(^EBi=^A)]TJ /F4 11.955 Tf 11.95 0 Td[(Ah(yi)]TJ /F9 11.955 Tf 12.25 0 Td[(y))]TJ /F10 11.955 Tf 11.95 9.68 Td[()]TJ /F9 11.955 Tf 7.18 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(XT^(S)i)]TJ /F4 11.955 Tf 11.96 0 Td[(A)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XT^(S))]TJ /F4 11.955 Tf 11.95 0 Td[(S+Ah)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(X)]TJ /F3 11.955 Tf 11.96 0 Td[(TS)]TJ /F10 11.955 Tf 11.96 9.69 Td[()]TJ /F9 11.955 Tf 5.76 -9.69 Td[(y)]TJ /F3 11.955 Tf 11.95 0 Td[(0)]TJ /F3 11.955 Tf 11.95 0 Td[(Ti=A1)]TJ /F4 11.955 Tf 11.95 0 Td[(A2+A3(say).Now, E)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(A23=A2h(S)TV)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(XS)]TJ /F9 11.955 Tf 11.96 0 Td[(2(S)TCov)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(X,y+V(y)i=1 mA2(S)T n+VS)]TJ /F9 11.955 Tf 11.95 0 Td[(2(S)TV+!=1 mA2(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g). (A) Nextbytheindependenceof(A1,A2)withA3,andtheexchangeabilityofthesequencenyi)]TJ /F9 11.955 Tf 12.25 0 Td[(y)]TJ /F10 11.955 Tf 11.95 9.68 Td[()]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XT^(S),Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(X,i=1,...,mo, E[(A1)]TJ /F4 11.955 Tf 11.96 0 Td[(A2)A3]=E(A1)]TJ /F4 11.955 Tf 11.96 0 Td[(A2)E(A3) (A) =1 mE(^A)]TJ /F4 11.955 Tf 11.96 0 Td[(AmXi=1h(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y))]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT^(S)i)E(A3))]TJ /F9 11.955 Tf 12.34 8.09 Td[(1 mE(AmXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(XT^(S))]TJ /F4 11.955 Tf 11.95 0 Td[(S)E(A3)=0. (A) Againusingexchangeability, E(A1A2)=1 mEf^A)]TJ /F4 11.955 Tf 11.96 0 Td[(AmXi=1h(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y))]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XT)]TJ /F9 11.955 Tf 21.35 3.15 Td[(^(S)T)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(X)]TJ /F9 11.955 Tf 14.65 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XTiA^(S))]TJ /F4 11.955 Tf 11.95 0 Td[(Sg=0, (A) 57

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wherethelastequationfollowssincemXi=1h(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y))]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT)]TJ /F9 11.955 Tf 21.36 3.16 Td[(^(S)T)]TJ /F9 11.955 Tf 7.18 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTi=SSByX)]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSB)]TJ /F6 7.97 Tf 6.58 0 Td[(1XSSByXTSSBX=0.Now, E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(A22=1 mA2E"^(S))]TJ /F4 11.955 Tf 11.96 0 Td[(STmXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XT^(S))]TJ /F4 11.955 Tf 11.96 0 Td[(S#=1 mA2E[)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XSSByXTSSBX)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XSSByX)]TJ /F9 11.955 Tf 9.3 0 Td[(2(S)TSSBX)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(SSB)]TJ /F6 7.97 Tf 6.59 0 Td[(1XSSByX+(S)TSSBXS]=1 mA2[E)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSBTyXSSB)]TJ /F6 7.97 Tf 6.58 0 Td[(1XSSByX)]TJ /F9 11.955 Tf 11.95 0 Td[(2(S)TE(SSByX)+(S)TE(SSBX)S]=1 mA2[(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)T(VS)+(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)p+(S)T(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1) n+VS)]TJ /F9 11.955 Tf 9.3 0 Td[(2(S)T(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)V+(S)T(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1) n+VS]=1 mA2(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g)p, (A) wherethepenultimateequalityfollowssinceE)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(SSBTyXSSB)]TJ /F6 7.97 Tf 6.58 0 Td[(1XSSByX=E[E(f[mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT]fmXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(X)]TJ /F9 11.955 Tf 14.65 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(XTg)]TJ /F6 7.97 Tf 6.58 0 Td[(1[mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(XT]gjX)]=Eftr[mXi=1)]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(X)]TJ /F9 11.955 Tf 14.65 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(XTg)]TJ /F6 7.97 Tf 6.58 0 Td[(1E ([mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(X][mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT])jX!]g 58

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=Eftr[mXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XTg)]TJ /F6 7.97 Tf 6.59 0 Td[(1[Var(mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XjX)+E(mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XjX)E(mXi=1yi)]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTjX)]]g=Eftr[mXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XTg)]TJ /F6 7.97 Tf 6.59 0 Td[(1[mXi=1Var(yijX))]TJ /F9 11.955 Tf 7.18 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.66 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT+mXi=1)]TJ /F9 11.955 Tf 7.18 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.65 Td[(XTS(S)TmXi=1)]TJ /F9 11.955 Tf 7.17 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(X)]TJ /F9 11.955 Tf 14.64 -7.03 Td[(Xi)]TJ /F9 11.955 Tf 13.64 2.65 Td[(XT]]g=(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)p+(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)(S)T n+VS.Finally,bytheindependenceofSSWywith(SSBy,SSBX,SSByX),SSWy2e2m(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1)and(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)jX(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)2m)]TJ /F8 7.97 Tf 6.59 0 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1,wehaveE)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(A21=1 mE"^A)]TJ /F4 11.955 Tf 11.95 0 Td[(A2mXi=1h(yi)]TJ /F9 11.955 Tf 12.24 0 Td[(y))]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F9 11.955 Tf 7.17 -7.02 Td[(Xi)]TJ /F9 11.955 Tf 13.65 2.66 Td[(XT^(S)i2#=1 mE^A)]TJ /F4 11.955 Tf 11.96 0 Td[(A2(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)=m)]TJ /F9 11.955 Tf 11.95 0 Td[(1 mE")]TJ /F9 11.955 Tf 5.88 -9.69 Td[(^2e=n2 ^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)]TJ /F9 11.955 Tf 11.95 0 Td[(2A)]TJ /F9 11.955 Tf 5.88 -9.68 Td[(^2e=n+A2(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)#=m)]TJ /F9 11.955 Tf 11.95 0 Td[(1 mE)]TJ /F9 11.955 Tf 5.88 -9.69 Td[(^2e=n2E1 ^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)]TJ /F9 11.955 Tf 11.96 0 Td[(2AE)]TJ /F9 11.955 Tf 5.88 -9.69 Td[(^2e=n+A2E(^!)]TJ /F9 11.955 Tf 12.2 0 Td[(^g)=m)]TJ /F9 11.955 Tf 11.95 0 Td[(1 m[)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(2e=n21+2 m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)=(m)]TJ /F4 11.955 Tf 11.95 0 Td[(p)]TJ /F9 11.955 Tf 11.95 0 Td[(3) (!)]TJ /F4 11.955 Tf 11.96 0 Td[(g))]TJ /F9 11.955 Tf 11.96 0 Td[(2A2e=n+A2(!)]TJ /F4 11.955 Tf 11.95 0 Td[(g)m)]TJ /F4 11.955 Tf 11.95 0 Td[(p)]TJ /F9 11.955 Tf 11.95 0 Td[(1 m)]TJ /F9 11.955 Tf 11.95 0 Td[(1]=A2(!)]TJ /F4 11.955 Tf 11.96 0 Td[(g)2n m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+o)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Combiningtheaboveresults,onegetsthetheorem. 59

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APPENDIXBDERIVATIONOFEQUATION3-24UsingtheformulaforthevarianceoftwoindependentvariablesandthefactthatA[n=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)](=n+)2m)]TJ /F6 7.97 Tf 6.58 0 Td[(1and[m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+2]^=Pi,j(zij)]TJ /F9 11.955 Tf 12.1 0 Td[(zi)22m(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)areindependent,onegetsV^B=V^(m)]TJ /F9 11.955 Tf 11.96 0 Td[(3)=A(m)]TJ /F9 11.955 Tf 11.95 0 Td[(1)=m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(12hE^i2V(1=A)+[E(1=A)]2V^+V^V(1=A)=m)]TJ /F9 11.955 Tf 11.95 0 Td[(3 m)]TJ /F9 11.955 Tf 11.95 0 Td[(12fm(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1) m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)+22[n=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)])]TJ /F6 7.97 Tf 6.58 0 Td[(2(=n+))]TJ /F6 7.97 Tf 6.58 0 Td[(22 (m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)2(m)]TJ /F9 11.955 Tf 11.96 0 Td[(5)+[n=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)])]TJ /F6 7.97 Tf 6.58 0 Td[(2(=n+))]TJ /F6 7.97 Tf 6.58 0 Td[(21 (m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)2 m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)+222m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)+ m(n)]TJ /F9 11.955 Tf 11.96 0 Td[(1)+222m(n)]TJ /F9 11.955 Tf 11.95 0 Td[(1)[n=(m)]TJ /F9 11.955 Tf 11.96 0 Td[(1)])]TJ /F6 7.97 Tf 6.58 0 Td[(2(=n+))]TJ /F6 7.97 Tf 6.58 0 Td[(22 (m)]TJ /F9 11.955 Tf 11.95 0 Td[(3)2(m)]TJ /F9 11.955 Tf 11.96 0 Td[(5)g=2B2 m1+1 n)]TJ /F9 11.955 Tf 11.95 0 Td[(1+o)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Notethatfortheaboveprooftowork,weneedm>5. 60

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BIOGRAPHICALSKETCH MeixiGuowasbornandraisedinBeijing,China.Shewasrecruitedtothegiftedchildren'sprogramofBeijingNo.8middleschoolin1997.In2002,shewasadmittedtotheSchoolofStatisticsandMathematicalScienceofBeijingNormalUniversity.In2006,sheearnedherbachelor'sdegreeinprobabilityandstatistics.Afterthat,shejoinedtheDepartmentofStatisticsattheUniversityofFlorida.Sheworkedasateachingassistantfortherstfouryearsandlaterbecameagraduateresearchassistantforanonprotorganizationwhosemissionistoservecollegesanduniversitiescommittedtoimprovinglearning,teaching,andleadershipperformance.SheisexpectedtoreceiveherdoctoratedegreeinAugust,2012. 65