SOME ALGEBRAIC AND STATISTICAL PROPERTIES OF WLSES

ELASOMEALGEBRAICANDSTATISTICALPROPERTIESOFWLSESUNDERAGENERALGROWTHCURVEMODEL∗YONGGETIAN†ANDYOSHIOTAKANE‡Abstract.Growthcurvemodelsareusedtoanalyzerepeatedmeasuresdata(longitudinaldata),whicharefunctionsoftime.Generalexpressionsofweightedleast-squaresestimators(WLSEs)ofparametermatricesweregivenunderageneralgrowthcurvemodel.Somealgebraicandstatisticalpropertiesoftheestimatorsarealsoderivedthroughthematrixrankmethod.AMSsubjectclassifications.62F11,62H12,15A03,15A09.Keywords.Growthcurvemodel,Moore-Penroseinverseofmatrix,Rankformulasformatrices,Parametermatrix,WLSE,Extremalrank,Uniqueness,Unbiasedness.1.Introduction.Throughoutthispaper,Rm×nstandsforthecollectionsofallm×nrealmatrices.ThesymbolsA,r(A),R(A)andtr(A)standforthetranspose,therank,therange(columnspace)andthetraceofamatrixA,respectively.TheKroneckerproductofanytwomatricesAandBisdefinedtobeA⊗B=(aijB).ThevecoperationofanymatrixA=[a1,...,an]isdefinedtobevec(A)=[a1,...,an].Awell-knownpropertyofthevecoperationofatriplematrixproductisvec(AZB)=(B⊗A)vec(Z).Alongitudinaldatasetisasetconsistingofagivensampleofindividualsovertime.Theyprovidemultipleobservationsoneachindividualsinthesample.Longitu-dinaldatacanbeusedtoestablishregressionmodelswithrespecttovariouspossibleregressors.Instatisticalapplications,aspecialtypeoflinearlongitudinaldatamodelisthefollowingwell-knowngrowthcurvemodelY=X1ΘX2+εεε,E(εεε)=0,Cov[vec(εεε)]=σ2(ΣΣΣ2⊗ΣΣΣ1).(1.1)ThemodelcanalsobewritteninthetripletformM={Y,X1ΘX2,σ2(ΣΣΣ2⊗ΣΣΣ1)},(1.2)whereY=(yij)∈Rn×misanobservablerandommatrix(alongitudinaldataset),X1=(x1ij)∈Rn×pandX2=(x2ij)∈Rq×maretwoknownmodelmatricesofarbitraryranks,Θ=(θij)∈Rp×qisamatrixofunknownparameterstobeestimated,ΣΣΣ1=(σ1ij)∈Rn×nandΣΣΣ2=(σ2ij)∈Rm×maretwoknownnonnegativedefinitematricesofarbitraryranks,and∗Receivedbytheeditors5March2007.Acceptedforpublication27July2007.HandlingEditor:RavindraB.Bapat.†SchoolofEconomics,ShanghaiUniversityofFinanceandEconomics,Shanghai200433,China([e-mail]).‡DepartmentofPsychology,McGillUniversity,Montr´eal,Qu´ebec,Canada([e-mail]).187ElectronicJournalofLinearAlgebraISSN1081-3810ApublicationoftheInternationalLinearAlgebraSocietyVolume16,pp.187-203,August2007σ2isapositiveunknownscalar.IfoneofΣΣΣ1andΣΣΣ2isasingularmatrix,(1.1)isalsosaidtobeasingulargrowthcurvemodel.ThroughtheKroneckerproductandthevecoperationofmatrices,themodelin(1.1)canalternativelybewrittenasvec(Y)=(X2⊗X1)vec(Θ)+vec(εεε),E[vec(εεε)]=0,Cov[vec(εεε)]=σ2(ΣΣΣ2⊗ΣΣΣ1).(1.3)Because(1.1)isalinearmodel,manyresultsonlinearmodelscarryoverto(1.1).Note,however,thatbothvec(Y)andvec(Θ)in(1.3)arevectors,andmanypropertiesofthetwomatricesYandΘin(1.1),suchastheirranks,range,singularity,symmetry,partitionedrepresentations,canhardlybedemonstratedintheexpressionsofvec(Y)andvec(Θ).Conversely,notallestimatorsofvec(Θ)and(X2⊗X1)vec(Θ)under(1.3)canbewrittenintheformsofmatricesin(1.1).Thatistosay,someproblemsonthemodel(1.1)canbestudiedthrough(1.3),otherscanonlybedonewiththeoriginalmodel(1.1).Thegrowthcurvemodelin(1.1)isanextensionofmultivariatelinearmodels.ThismodelwasoriginallyproposedbyPotthoffandRoy[11]instudyinglongitudinaldataandwassubsequentlystudiedbymanyauthors,suchas,Frees[4],Hsiao[5],Khatri[7],PanandFang[9],Rao[14,15],Seber[18],vonRosen[31,32],andWoolsonandLeeper[33],amongmanyothers.Thepurposeofthepresentpaperistogivesomegeneralexpressionsofweightedleast-squaresestimators(WLSEs)ofΘ,X1ΘX2andK1ΘK2underthegeneralassumptionin(1.1),andthenstudythemaximalandminimalpossibleranksoftheestimators,aswellastheunbiasednessandtheuniquenessoftheestimators.TheMoore-PenroseinverseofA∈Rm×n,denotedbyA+,isdefinedtobetheuniquesolutionGtothefourmatrixequations(i)AGA=A,(ii)GAG=G,(iii)(AG)=AG,(iv)(GA)=GA.AmatrixGiscalledageneralizedinverse(g-inverse)ofA,denotedbyG=A−,ifitsatisfies(i).Further,letPA,FAandEAstandforthethreeorthogonalprojectorsPA=AA+,EA=Im−AA+andFA=In−A+A.InordertosimplifyvariousmatrixexpressionsconsistingoftheMoore-Penroseinversesofmatrices,weneedsomeformulasforranksofmatrices.ThefollowingrankformulasareduetoMarsagliaandStyan[8].Lemma1.1.LetA∈Rm×n,B∈Rm×k,C∈Rl×nandD∈Rl×k.Thenr[A,B]=r(A)+r(EAB)=r(B)+r(EBA),(1.4)r
AC=r(A)+r(CFA)=r(C)+r(AFC),(1.5)r
ABC0=r(B)+r(C)+r(EBAFC).(1.6)ElectronicJournalofLinearAlgebraISSN1081-3810ApublicationoftheInternationalLinearAlgebraSocietyVolume16,pp.187-203,August2007(B)⊆R(A)andR(C)⊆R(A),thenr
ABCD=r(A)+r(D−CA+B).(1.7)Lemma1.2.SupposeR(B1)⊆R(C1),R(B2)⊆R(C1),R(B2)⊆R(C2)andR(B3)⊆R(C2).Thenr(A−B1C+1B2C+2B3)=r0C2B3C1B20B10−A−r(C1)−r(C2).(1.8)Proof.IntermsoftheMoore-Penroseinversesofmatrices,thegivenconditionsareequivalenttoB1C+1C1=B1,C1C+1B2=B2,B2C+2C2=B2,C2C+2B3=B3.(1.9)Alsorecallthatelementaryblockmatrixoperations(EBMOs)donotchangetherankofamatrix.Applying(1.9)andEBMOstotheblockmatrixin(1.8)givesr0C2B3C1B20B10−A=r0C2B3C1000−B1C+1B2−A=r0C20C10000−A+B1C+1B2C+2B3=r(A−B1C+1B2C+2B3)+r(C1)+r(C2),establishing(1.8).Let