Asymptotic theorems for urn models with nonhomogen

StochasticProcessesandtheirApplications80(1999)87{101AsymptotictheoremsforurnmodelswithnonhomogeneousgeneratingmatricesZ.D.Bai,FeifangHuDepartmentofStatisticsandAppliedProbability,NationalUniversityofSingapore,Singapore119260,SingaporeReceived11February1998;receivedinrevisedform29October1998;accepted30October1998AbstractThegeneralizedFriedman'surn(GFU)modelhasbeenextensivelyappliedtobiostatistics.However,intheliterature,alltheasymptoticresultsconcerningtheGFUareestablishedun-dertheassumptionofahomogeneousgeneratingmatrix,whereas,inpracticalapplications,thegeneratingmatricesareoftennonhomogeneous.Ontheotherhand,evenforthehomogeneouscase,thegeneratingmatrixisassumedintheliteraturetohaveadiagonalJordanformandsatises2Re(1),whereand1arethelargesteigenvalueandtheeigenvalueofthesec-ondlargestrealpartofthegeneratingmatrix(seeSmythe,1996,StochasticProcess.Appl.65,115{137).Inthispaper,westudytheasymptoticpropertiesoftheGFUmodelassociatedwithnonhomogeneousgeneratingmatrices.Theresultsareapplicabletoavarietyofsettings,suchastheadaptiveallocationruleswithtimetrendsinclinicaltrialsandthosewithcovariates.TheseresultsalsoapplytothecaseofahomogeneousgeneratingmatrixwithageneralJor-danformaswellasthecasewhere=2Re(1).c1999ElsevierScienceB.V.Allrightsreserved.AMSclassication:primary62E20;62L05;secondary62F12Keywords:Adaptivedesigns;Asymptoticnormality;Consistency;GeneralizedFriedman'surnmodel;Non-homogeneousgeneratingmatrix1.IntroductionAdaptivedesignshaveoftenbeenproposedasawaysequentiallytoassignmorepatientstobettertreatments,basedonoutcomesofprevioustreatmentsinclinicaltrials.AveryimportantclassofadaptivedesignsisonebasedonthegeneralizedFriedman'surn(GFU)model(seeAthreyaandKarlin(1968);GFUisalsonamedasgeneralizedPolyaurn(GPU)intheliterature),whichhasapplicationinclinicaltrials,bioassayandpsychophysics.ReferencesaremadetoWei(1979),Rosenbergeretal.(1997)andCorrespondingauthor.0304-4149/99/${seefrontmatterc1999ElsevierScienceB.V.Allrightsreserved.PII:S0304-4149(98)00094-588Z.D.Bai,F.Hu/StochasticProcessesandtheirApplications80(1999)87{101RosenbergerandGrill(1997).AgeneralreviewonthissubjectwithrespecttoclinicaltrialsisgiveninRosenberger(1996).AdaptivedesignsusingtheGFUmodelcanbeformulatedasfollows.Assume,atthebeginning,anurncontainsparticlesofKdistincttypes,denotedbyY0=(Y01;:::;Y0K),respectivelyrepresentingK`treatments'inaclinicaltrial,whereY0kdenotesthenumberofparticlesoftypek,k=1;:::;K.Thesetreatmentsaretobesequentiallyallocatedinnconsecutivestages.Atstagei,i=1;:::;naparticleisdrawnfromtheurnwithreplace-ment.Ifatypekparticleisdrawnattheithstage,thenthetreatmentkisassignedtothepatienti,k=1;:::;K;i=1;:::;n.Let(i)denotearandomvariableassociatedwiththeithstageoftheclinicaltrial,whichmayincludemeasurementsontheithpatientandtheoutcomeofthetreatmentattheithstage.Afterwards,additionalDk;q(i)particlesoftypeqareaddedtotheurn,q=1;:::;K,whereDk;q(i)isafunctionof(i).Thisprocedureisrepeatedtothenthstage.Afternsplitsandgenerations,thecompositionoftheurnisdenotedbythevectorYn=(Yn1;:::;Ynk),whereYnkrepresentsthenumberoftypekparticlesintheurn.Furthermore,wedeneDi=hhDk;q(i);k;q=1;:::;KiiandHi=hhE(Dk;q(i));k;q=1;:::;Kii,i=1;:::;n.ThematrixDi'sarecalledrulesandHi'sarethegeneratingmatrices.WecalltheGFUmodelhomogeneousifHi=Hforalli=1;:::;n.Forahomoge-neousGFUmodel,undertheassumptions(i)PrfDk;q=0;q=1;:::;Kg=0foreveryk=1;:::;Kand(ii)Hispositiveregular,AthreyaandKarlin(1968)andAthreyaandNey(1972)showthatNnkn!vkandYnkPKq=1Ynq!vk(1.1)almostsurelyasn!1,wherev=(v1;:::;vK)isthelefteigenvectorofthelargesteigenvalueofH.Let1denotetheeigenvalueofthesecondlargestrealpart,withcorrespondingrighteigenvector.Furthermore,undertheadditionalassumptionthat2Re(1),AthreyaandKarlin(1968)showthatn−1=2Yn0!N(0;2);(1.2)where2isaconstant.When=2Re(1)and1isasimpleeigenvalue,then(1:2)holdswiththenormalizationconstantpnreplacedbyofpnlnn:Smythe(1996)denestheExtendedPolyaUrnmodel(EPU)asaGFUwithPKq=1E(Dk;q)=c0;k=1;:::;K,namely,addinganexpectedconstanttotalnum-berofballsateachstage.FortheEPU,Smythe(1996)establishedtheasymptoticnormalityofYnandNnundertheassumptions:(i)foreachnonprincipaleigenvaluej;2Re(j);(ii)alleigenvaluesaresimple,andnotwodistinctcomplexeigen-valueshavethesamerealpart,exceptforconjugatepairs;and(iii)theeigenvectorsarelinearlyindependent,whereNn=(Nn1;:::;NnK)andNnKisthenumberoftimesatypekparticledrawnintherstntrials.Inthispaper,theasymptoticpropertiesoftheurncompositionYn=(Yn1;:::;YnK)areinvestigatedforEPUswithnonhomogeneousgeneratingmatricesfHig.Throughoutthispaper,weassumePKq=1Dkq(i)=c0,forallk=1;:::;Kandi=1;:::;n,i.e.,addingatotalnumberofballsateachstage.WealsoassumethatthereexistsapositiveZ.D.Bai,F.Hu/StochasticProcessesandtheirApplications80(1999)87{10189regularmatrixHsuchthat1Xi=1ii1;(1.3)wherei=kHi−Hk1.Theorems2.1and2.2showthatKXi=1Yni!−1(Yn1;:::;YnK)TconvergestothelefteigenvectorcorrespondingtothemaximaleigenvalueofHinprobability.FurtherweshowtheasymptoticnormalityofYninTheorem2.3.Toillustratethatthetheoryofnonhomogeneou