CHAPTER11GeneralGroupSequentialDistributionTheory11.1IntroductionInChapter3weshowedthatawidevarietyofsituationsanddatatypesleadtosequencesofstatisticswithacommonformofjointdistribution.Consequently,thegroupsequentialmethodspresentedinChapters2to10arewidelyapplicable.Inthischapterweprovegeneralresultswhichbroadenfurthertheapplicabilityofthesemethods.Ifobservationsfollowanormallinearmodel,thedistributiontheoryforasequenceofestimatesisexact.WederivethistheoryinSection11.3andshowthatitextendstocorrelatedobservations,suchasrepeatedmeasurementsonsubjectsinalongitudinalstudy,aslongasthecorrelationstructureisknown.Ifthecovariancematrixinanormallinearmodeliscompletelyknown,Z-statisticscanbecreatedandthesewillhavethecanonicaljointdistribution(3.1).Inmostapplications,avarianceparameterσ2mustbeestimatedfromtheobserveddata.Groupsequentialtestsmaythenbebasedonasequenceoft-statisticsandapproximateproceduresfordoingthishavebeenpresentedinChapters2to6.ResultspresentedinSection11.4showthatthejointdistributionsofsequencesoft-statisticsconformtoastandardpatternandexactgroupsequentialmethodscanbebasedonthisjointdistribution.Fornon-normaldata,thetheoryisasymptoticassamplesizeincreases,butitprovidesagoodapproximationinsmallsamplesandthiscanserveasthebasisofgroupsequentialtestsforparametersin,forexample,ageneralizedlinearmodel.WehavealreadyseensuchapplicationsinSections3.5,3.6and10.5.ThetheoryforgeneralparametricmodelsispresentedinSection11.6.Statisticsarisinginthegroupsequentialanalysisofsurvivaldatasharesimilarasymptoticproperties.Theoryforthelog-ranktest,whichunderliestheexampleinSection3.7,andforsemi-parametricregressionmodelsforsurvivaldatasubjecttocensoringwillbediscussedinChapter13.11.2AStandardJointDistributionforSuccessiveEstimatesofaParameterVectorConsideragroupsequentialstudywithamaximumofKgroupsofobservations.Let�β(k),k=1,...,K,denotetheestimatesoftheparametervectorβ=(β1,...,βp)Tatsuccessiveanalyses.Itisnotuncommontofind,atleastc⃝2000byChapman&Hall/CRCapproximately,th...
