CHAPTER4One-SidedTests4.1IntroductionThenameone-sidedtestisgiventoatestofahypothesisagainstaone-sidedalternative,forexample,atestofthenullhypothesisH0:θ=0againstthealternativeHA:θ>0.IfafixedsampletestbasedonastatisticZrejectsH0whenZ>c,forsomeconstantc,thetest’sTypeIerrorprobabilityisPrθ=0{Z>c},anditspoweristheprobabilityofrejectingH0undervaluesofθinthealternativehypothesis,i.e.,Prθ{Z>c}forvaluesθ>0.Insomeproblems,aone-sidedalternativeisappropriatebecausedeparturesfromH0intheotherdirectionareeitherimplausibleorimpossible.Inothercases,θmaylieoneithersideofH0butthealternativeischosenintheonedirectioninwhichdeparturesfromH0areofinterest.Supposeθmeasurestheimprovementinresponseachievedbyanewtreatmentoverastandard,andthenewtreatmentwillonlybedeemedacceptableifitproducessuperiorresponses;thenthemainaimofastudywillbetodistinguishbetweenthecasesθ≤0andθ>0.However,theprobabilityofdecidinginfavorofthenewtreatmentwhenθ≤0isgreatestatθ=0,soitsufficestospecifytheTypeIerroratθ=0.Itmatterslittlewhetheroneregardsthenullhypothesisinsuchastudyasθ≤0orθ=0.WeadoptthelatterformulationandsodiscusstestsofH0:θ=0againstHA:θ>0designedtoachieveTypeIerrorprobabilityαwhenθ=0andpower1−βataspecifiedalternativevalueθ=δwhereδ>0.Weshallpresentgroupsequentialone-sidedtestsforaparameterθwhenstandardizedstatisticsZ1,...,ZKavailableatanalysesk=1,...,Kfollowthecanonicaljointdistribution(3.1).InparticularZk∼N(θ√Ik,1),whereIkistheinformationforθatanalysisk.AsexplainedinChapter3,thisjointdistributionarises,sometimesapproximately,inagreatmanyapplications.Thegroupsequentialone-sidedtestsweshalldefinearedirectlyapplicabletoallthesesituations.AfixedsampletestbasedonastatisticZdistributedasN(θ√I,1)attainsTypeIerrorprobabilityαbyrejectingH0ifZ>�−1(1−α).Hence,inordertoattainpower1−βwhenθ=δ,thesamplesizemustbechosentogiveaninformationlevelIf,1={�−1(1−α)+�−1(1−β)}2/δ2,(4.1)thesubscript1inIf,1denotingaone-sidedtest.Asbefore,�denotesthestandardnormalcdf.Asfortwo-sid...
