Estimating Features of a Distribution From Binomia

ESTIMATINGFEATURESOFADISTRIBUTIONFROMBINOMIALDATA
ArthurLewbelBostonCollegeOliverLintonLondonSchoolofEconomicsDanielMcFaddenUniversityofCalifornia,BerkeleyDecember5,2001AbstractAstatisticalproblemthatarisesinseveralÞeldsisthatofestimatingthefeaturesofanunknowndis-tribution,whichmaybeconditionedoncovariates,usingasampleofbinomialobservationsonwhetherdrawsfromthisdistributionexceedthresholdlevelssetbyexperimentaldesign.Oneapplicationisde-structivedurationanalysis,wheretheprocessiscensoredatanobservationtesttime.Anotherisreferen-dumcontingentvaluationinresourceeconomics,whereoneisinterestedinfeaturesofthedistributionofvaluesplacedbyconsumersonapublicgoodsuchasendangeredspecies.SampleconsumersareaskedwhethertheywouldvoteforareferendumthatwouldprovidethegoodatacostspeciÞedbyexperimen-taldesign.Thispaperprovidesestimatorsformomentsandquantilesoftheunknowndistributioninthisproblem.
ThisresearchwassupportedinpartbytheNationalScienceFoundationthroughgrantsSES-9905010andSBR-9730282,andbytheE.MorrisCoxEndowment.DepartmentofEconomics,BostonCollege,140CommonwealthAvenue,ChestnutHill,MA02467,USA.Phone:[phone].E-mailaddress:[e-mail],LondonSchoolofEconomics,HoughtonStreet,LondonWC2A2AE,UnitedKingdom.E-mailaddress:[e-mail],UniversityofCalifornia,Berkeley,CA94720-3880,USA.E-mailaddress:[e-mail]11IntroductionAstatisticalproblemthatarisesinseveralÞeldsisthatofestimatingthefeaturesofanunknowndistribu-tion,whichmaybeconditionedoncovariates,usingasampleofbinomialobservationsonwhetherdrawsfromthisdistributionexceedthresholdlevelssetbyexperimentaldesign.Threeapplicationsillustratetheproblem:Bioassay-Findthedistributionofsurvivaltimesuntiltheonsetofanabnormalityinlaboratoryanimalsexposedtoanenvironmentalhazard.TheanimalsaresacriÞcedattimesdeterminedbyexperimentaldesign,andtestedfortheabnormality.Anobservationconsistsofavectorofcovariates,atesttime,andanindicatorforthetestresult.DestructiveTesting-Findthedistributionofspeedsatwhichairbagsfailtoprotectpassengersinautomobilecrashes.Atspeedsselectedbyexperimentaldesign,drivecarsintoabarrieranddeterminewhetheradummyoccupantisinjured.Anobservationconsistsofcovariates,atestspeed,andanindicatorforinjury.SurveyresearchwithShadowEffects-Findthedistributionofahouseholdeconomicvariablesuchaswealth.Subjectsareaskediftheireconomicvariableexceedsatestvaluechosenbydesign.Anobservationconsistsofcovariates,atestvalue,anindicatorfortheresponse.FollowupqueriesareshadowedbytheframingeffectoftheÞrstbid.Thisshadowingeffectiscommoninunfoldingbracketsurveyquestionsoneconomicvariables,andonstatedwillingnesstopay(WTP)foreconomicgoods.1Givenasetofcovariates,whentheexperimentaldesignisrandomizedwithastrictlypositivetestvaluedensityandmildregularityconditions,weproposeconsistentestimatorsforconditional(oncovariates)momentsoftheunknowndistribution.Wealsoproviderootnconsistentestimatorsforthecasewheretheunknowndistributiondependsoncovariatesthroughasingleindexlocationshift.Inaddition,weprovideestimatorsofconditionalquantilesoftheunknowndistribution.1McFadden(1994)providesreferencesandexperminatalevidencethatresponsestofollowuptestvaluescanbebiased.Thereareadditionalissuesoftheimpactofframingofquestionsonsurveyresponses,particularlyanchoringtotestvalues,includingtheinitialtestvalue;seeGreenetal.(1998)andHurdetal.(1998).Thedatagenerationprocessmaythenbeaconvolutionofthetargetdistributionandadistributionofpsychometricerrors.Thispaperwillignoretheseissuesandtreatthedatagenerationprocessasifitisthetargetdistribution.ThedifÞcultproblemofdeconvolutingatargetdistributioninthepresenceofpsychometricerrorsisleftforfutureresearch.22ModelSpeciÞcationThegoalisestimationofconditionalmomentsorquantilesofalatent,unobservedrandomscalarW,condi-tionedonavectorofobservedcovariatesX.TheconditionalcumulativedistributionfunctionofW,denotedG

x,isunknownbutassumedtobesmooth.Atestvalueissetbyarandomizedexperimentaldesignornaturalexperiment.ThevalueisarealizationofarandomvariableVdrawnfromeitheraknownorunknownconditionaldensityh

x(weconsiderbothcases).ItisassumedthatWisconditionallyindependentofV,conditioningonX(consistentwithexperimentaldesign).DeÞneYtoequaloneintheeventthatWexceedsV,andzerootherwise,soYI
WVwhereI
istheindicatorfunction.TheobserveddataconsistofarandomsampleofrealizationsofcovariatesX,testvaluesV,andoutcomesY.Theframeworkissimilartorandomcensoredregressions(withcensoringpoint),exceptthatforrandomcensoringwewouldobserveforobservationshaving,whereasinthepresentcontextweonlyobserveyI
.Givenafunctionr
x,thegoalisestimationoftheconditionalmomentr
xE[r
WX
Xx]foranychosenxinthesupportofX.Ofparticularinterestarethemomentsbasedonr
WXWkforintegerskInadditiontomomentswemayalsobeinterestedinquantiles.Letq
xdenotetheq’thquantileofWgivenxIftheconditionaldistributionofWgivenXxisÞnitelyparameterized,thenthoseparameterscangenerallybeefÞcientlyestimatedbymaximumlikelihood(correspondingtoordinarybinarychoicemodelestimation,