Constancy of distributions asymptotic efficiency o

Constancyofdistributions:asymptoticefficiencyofcertainnonparametrictestsofconstancyAlexJ.KoningEconometricInstituteErasmusUniv[e-mail]ofMathematicsandSt[e-mail]portEI2002-33AbstractInthispaperwestudystochasticprocesseswhichenablemonitoringthepos-siblechangesofprobabilitydistributionsovertime.Theseso-calledmonitoringprocessesarebivariatefunctionsoftimeandpositionatthemeasurementscale,andinparticularbeusedtotestthenullhypothesisofnochange:onemaythenformKolmogorov–Smirnovorothertypeoftestsasfunctionalsoftheprocesses.InHjortandKoning(2001)Cram´er-typedeviationresultswereobtainedundertheconstancynullhypothesisfor[bootstrappedversionsof]such“derived”teststatistics.Herethebehaviourofderivedteststatisticsisinvestigatedunderalternativesinthevicinityoftheconstancyhypothesis.WhencombinedwithCram´er-typedeviationresults,theresultsinthispaperenablethecomputationofefficienciesofthecorrespondingtests.Thediscussionofsomeexamplesofyieldguidelinesforthechoiceoftheteststatistic,andhencefortheunderlyingmonitoringprocess.1IntroductionandsummaryAssumethatindependentdataareavailableforeachofconsecutiveoccasions,per-hapsmeasurementsofsomequantitytakenonseparatedates.Thenullhypothesistobetestedhereisthatof(1)whereisthecumulativedistributionfunctionspecifyingthedistributionofdataonoccasion.Weshallrefertoasthesubsample.Together,thesubsamplesformthefullsample.1Constancyofdistributions:asymptoticefficiency2Onemaythinkof(1)asthehypothesisthataninfinitedimensionalparameterremainsconstant.Inthisperspective,isthevalueofinthe!subsample.Weshalldenotethesize$#%’&&&%(*)ofthefullsampleby.Althoughitisnotreflectedinnotation,notethatdependson+,andtendstoinfinityas+tendstoinfinity.Thesubsamplesizesareallowedtoberandom,andareconvenientlyrepresentedbytherandomprobabilitymeasure,)-./0$1#2)345#67.98:;7=&Undertheassumptionthat,)-./convergestoadeterministicfunction,-./insomepredescribedmanneras+$@?,nullhypothesistheoryforstochasticprocesseswhichenablemonitoring(1)ispresentedinHjortandKoning(2001);inparticular,Koml´os-Major-Tusn´adytypeinequalitiesareemployedtoobtaindeviationresultsfor[boot-strappedversionsof]teststatisticsbasedonthesemonitoringprocesses.Inthesequelweshallrefertothesestatisticsas“derivedteststatistics”.Inthispaperwedevelop“localalternativestheory”;thatis,theoryforthebe-haviourofthemonitoringprocessesunderalternativesinthevicinityofthenullhy-pothesis.IncombinationwithnullhypothesistheoryasinHjortandKoning(2001),thelocalalternativestheoryenablesustoassesstheabilityofamonitoringprocesstodetectdeparturesfromthenullhypothesis.Infact,weshallinvestigatetheper-formanceofaderivedteststatisticbyevaluatingvarious“local”efficiencymeasureswhichpertaintothebehaviourofthepowercurveinthevicinityofthenullhypothesis.Localefficiencymeasuresaretypicallyusedasaselectiondeviceforstatisticaltests,asforanytwotestswhichdifferinefficiencythereisavicinityofthenullhy-pothesisinwhichthemoreefficientoneismorepowerfulthanthelessefficientone.Foranenthusiasticreviewoftheroleofefficiencymeasuresinthedevelopmentofnonparametricstatistics,werefertoNikitin(1995).Inordertocomputethevariouslocalefficienciesinaunifiedmanner,wefirstshowthatthederivedteststatisticsatisfiesConditionIIIAinWieand(1976)[cf.Defi-nition2(c)inSection3].ThecombinationofamoderatedeviationresultunderthenullhypothesisandConditionIIIAinthevicinityofthenullhypothesisenablesthecom-putationoflimiting[asthealternativeapproachesthenullhypothesis]approximateBahadurefficiency,limiting[asthesizeofthetesttendstozero]Pitmanefficiency,andweakasymptotici-efficiency.Moreover,replacingthemoderatedeviationresultbyaCram´ertypedeviationresult[Chernofftypedeviationresult]yieldsasymptotici-efficiency[strongasymptotici-efficiency].Followinginspectionofthestructureoftheevaluatedefficiency,weformulateguidelinesforconstructinghighlyefficientderivedtests.Indirectly,theseguidelinesshedlightontheperformanceoftheunderlyingmonitoringprocessaswell.Theoutlineofthepaperisasfollows.InSection2weintroducethemonitoringprocesses,andstudytheirbehaviourunderalternativesinthevicinityofthenullhy-pothesis.InSection3weusetheresultsofSection2tocomputelocalefficienciesofConstancyofdistributions:asymptoticefficiency3derivedteststatistics.InSection4themethodsareappliedtoseawaterleveldata.ProofsaregatheredinSection5.2Thealternativehypothesis2.1NotationandpreliminariesInthissectionweassumethatthecumulativedistributionfunctionsBCDEEEDBGFarenotequaltoeachother,butinsteadcoincidewiththeHJIKrowBFLCMNOPLQJDEEEDBGFFJMNOPLQofatriangulararrayindexedbyPSRUT.LetVtheclassofallprobabilitymeasuresunderconsideration.ItisconvenienttothinkofBFWMNOPXQasthecumulativedistributionfunctionbelongingtotheYIKsubsampleatstageHundertheprobabilitymeasureZ[9\V^]V9_,wherePindicatesthedistanceofthealternativetothenullhypothesis.Thenullhypothesisisassum