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1、SequentialEstimationbyMalayGhosh,NitisMukhopadhyayandPranabK.SenCopyright©1997JohnWiley&Sons,Inc.Chapter12Time-SequentialEstimationProblems12.1INTRODUCTIONInmedicalstudiesandlife-testingproblems,generally,observationsaregath-eredsequentiallyovertime.Typicallyoneencountersasetofnonneg
2、ativerandomvariableswithadistributionfunction(ofpossiblyunspecifiedfunc-tionalform)definedon1R+,wheretheexperimentaloutcomerelatestothesuccessiveorderstatisticsoccurringsequentiallyovertime,andtheremaybeotherplausibleconcomitantvariables.Further,duetotimeandcostcon-straintsonthestudy
3、,theremaybesomeformof(i.e.,TypeI,TypeII,orrandom)censoringthatmayinducefurthercomplicationsinstatisticalmodelingandanalysis,whileduetootherplausibleethicalfactors,anearlystoppingmaybeadvocated,oftenonnonstatisticalconsiderationstoo.Thustherearestructuraldifferencesbetweentheclassical
4、sequentialestimationproblemstreatedearlierandtheirtime-sequentialcounterpartstobetreatedhere.Intheclassicalsequentialpointaswellasintervalestimationproblems,thesamplesize(beinggovernedbyawell-definedstoppingrule)isanonnegativeinteger-valuedr.v.Thoughgatheredsequentially,thesampleobse
5、rvationsmayberecordedinstantaneously;thatis,theydonotincuranyfollow-uptime,soonehasessentiallyarandomnumberofi.i.d.r.v.'swhichareincorpo-ratedintheformulationofstoppingruleandestimationrule.Ontheotherhand,inatypicalfollow-upstudy,thesampleobservations,asgatheredovertime,relatetotheti
6、metofailureoroccurrencesofsomecharacteristiceventsalongwithotherplausibleconcomitantvariables.Thoughcloselyrelatedtothesampleorderstatistics,theseobservationsaregenerallyneitherindepen-dentnoridenticallydistributed.Itisalsonotuncommontohaveastatisticalmonitoringprocedurefromthebeginn
7、ing,onamoreorlesscontinualbasis,sothatthestoppingtime,referringactuallytothetime-pointwhenexper-imentationiscurtailed,maynotbeaninteger-valuedr.v.Thusonemay34334412.TIME-SEQUENTIALESTIMATIONPROBLEMSencounteramoregeneralsituationwherethestoppingruleisformulatedonthebasisoftheaccumulat
8、ingdataovert
