CHAPTER10StochasticCurtailment10.1IntroductionThisapproachtosequentialmonitoringevolvedfromtheideaofsimplecurtailmentwherebyanexperimentcanbeterminatedassoonastheultimateresultbecomesinevitable.Asanexample,consideranindustrialacceptancesamplingschemeinwhichitemsareclassifiedaseitherdefectiveornon-defective.Asingle-samplingschemedesignatesalotasacceptableiffewerthancdefectivesarefoundinarandomsampleofnitemstakenfromthelot.Inthecurtailedversionofthisprocedure(e.g.,Alling,1966),samplingproceedssequentiallyandterminatesassoonasthedecisionisinevitable,thatis,wheneitherthenumberofdefectivesreachescorthenumberofnon-defectivesreachesn−c+1.Clearly,thiscurtailedschemealwaysleadstothesameconclusionasthefixedsampleprocedure.Thetwomethodshaveidenticaloperatingcharacteristics,butthecurtailedprocedurehasalowerexpectedsamplesize.Anyfixedsampletestwithboundedcontributionsfromeachnewobservation,suchastheWilcoxonranksumtestormanyothernonparametrictests,canbecurtailedinasimilarway;seeAlling(1963)andHalperin&Ware(1974).Ifoutcomemeasuresarenotbounded,asinthecaseofnormallydistributedobservations,orifgreatersavingsinexpectedsamplesizearedesired,thenastochasticformofcurtailmentmaybeconsidered.Atanystagewemightterminatetheprocedureifaparticulardecision,whilenotinevitable,ishighlylikelygiventhecurrentdata.Wecallsucharuleastochasticallycurtailedprocedure(SCP).Jennison(1992)reportsthesuccessofsuchrulesinareducingcomputationinasimulationstudyconductedtoevaluatetheactualerrorratesofabootstraptest.Aformofstochasticcurtailmentimprovedgreatlyonsimplecurtailment,andcomputationinthisapplicationwasreduced,overall,byafactorofasmuchas50.InthefollowingSections,wedescribethreeapproachestosuchstochasticcurtailment,theconditionalpowerapproach,thepredictivepowerapproachandaparameterfreeapproach.Weillustratethesemethodsinacasestudyinvolvingsurvivaldata.10.2TheConditionalPowerApproachSupposewewishtotestanullhypothesisH0:θ=θ0concerningtheparameterθ.Firsta“referencetest”,whichweshallcallT,isdesignated.Th...
