Local polynomial regression estimators in survey s

LocalPolynomialRegressionEstimatorsinSurveySamplingF.JayBreidtandJeanD.OpsomerIowaStateUniversityJune8,2000AbstractEstimationofnitepopulationtotalsinthepresenceofauxiliaryinformationisconsidered.Aclassofestimatorsbasedonlocalpolynomialregressionisproposed.Likegeneralizedregressionestimators,theseestimatorsareweightedlinearcom-binationsofstudyvariables,inwhichtheweightsarecalibratedtoknowncontroltotals,buttheassumptionsonthesuperpopulationmodelareconsiderablyweaker.Theestimatorsareshowntobeasymptoticallydesign-unbiasedandconsistentun-dermildassumptions.AvarianceapproximationbasedonTaylorlinearizationissuggestedandshowntobeconsistentforthedesignmeansquarederroroftheestimators.TheestimatorsarerobustinthesenseofasymptoticallyattainingtheGodambe-Joshilowerboundtotheanticipatedvariance.Simulationexperimentsindicatethattheestimatorsaremoreecientthanregressionestimatorswhenthemodelregressionfunctionisincorrectlyspecied,whilebeingapproximatelyasecientwhentheparametricspecicationiscorrect.AMS2000subjectclassication.Primary62D05;secondary62G08.Keywordsandphrases.Calibration,generalizedregressionestimation,Godambe-Joshilowerbound,model-assistedestimation,nonparametricregression.IowaStateUniversityofScienceandTechnology,StatisticalLaboratoryandDepartmentofStatistics,221SnedecorHall,Ames,IA50011{1210.LocalSurveyRegressionEstimators11Introduction1.1BackgroundInmanysurveyproblems,auxiliaryinformationisavailableforallelementsofthepopu-lationofinterest.Populationregistersinsomecountriescontainageandtaxableincomeforallresidents.Studiesoflaborforcecharacteristicsorhouseholdexpenditurepatternsmightbenetfromtheseauxiliarydata.Geographicinformationsystemsmaycontainmea-surementsderivedfromsatelliteimageryforalllocations.Thesespatially-explicitdatacanbeusefulinaugmentingmeasurementsobtainedinagriculturalsurveysornaturalre-sourceinventories.Indeed,useofauxiliaryinformationinestimatingparametersofanitepopulationofstudyvariablesisacentralprobleminsurveys.Oneapproachtothisproblemisthesuperpopulationapproach,inwhichaworkingmodeldescribingtherelationshipbetweentheauxiliaryvariablexandthestudyvari-ableyisassumed.Estimatorsaresoughtwhichhavegoodeciencyifthemodelistrue,butmaintaindesirablepropertieslikeasymptoticdesignunbiasedness(unbiasednessoverrepeatedsamplingfromthenitepopulation)anddesignconsistencyifthemodelisfalse.Typically,theassumedmodelsarelinearmodels,leadingtothefamiliarratioandregressionestimators(e.g.,Cochran,1977),thebestlinearunbiasedestimators(Brewer,1963;Royall,1970),thegeneralizedregressionestimators(Cassel,Sarndal,andWretman,1977;Sarndal,1980;RobinsonandSarndal,1983),andrelatedestimators(Wright,1983;IsakiandFuller,1982).Thepaperscitedvaryintheiremphasisondesignandmodel,butitisfairtosaythatallareconcernedtosomeextentwithbehavioroftheestimatorsundermodelmisspecication.Giventhisconcernwithrobustness,itisnaturaltoconsideranonparametricclassofmodelsfor,becausetheyallowthemodelstobecorrectlyspeciedformuchlargerclassesoffunctions.Kuo(1988),Dorfman(1992),DorfmanandHall(1993),andChambers,Dorfman,andWehrly(1993)haveadoptedthisapproachinconstructingmodel-basedestimators.Thispaperdescribestheoreticalpropertiesofanewtypeofmodel-assistednonparamet-ricregressionestimatorforthenitepopulationtotal,basedonlocalpolynomialsmoothing.Localpolynomialregressionisageneralizationofkernelregression.Cleveland(1979)andClevelandandDevlin(1988)showedthatthesetechniquesareapplicabletoawiderangeofproblems.TheoreticalworkbyFan(1992,1993)andRuppertandWand(1994)showedthatithasmanydesirabletheoreticalproperties,includingadaptationtothedesignofthecovariate(s),consistencyandasymptoticunbiasedness.WandandJones(1995)provideaclearexplanationoftheasymptotictheoryforkernelregressionandlocalpolynomialregres-sion.ThemonographbyFanandGijbels(1996)exploresawiderangeofapplicationareasoflocalpolynomialregressiontechniques.However,theapplicationofthesetechniquestomodel-assistedsurveysamplingisnew.InSection1.2weintroducethelocalpolynomialregressionestimatorandinSection1.3westateassumptionsusedinthetheoreticalderivationsofSection2,inwhichourmainresultsaredescribed.Section2.1showsthattheestimatorisaweightedlinearcombinationofstudyvariablesinwhichtheweightsarecalibratedtoknowncontroltotals.Section2.2containsaproofthattheestimatorisasymptoticallydesignunbiasedanddesignconsis-tent,andSection2.3providesanapproximationtoitsmeansquarederrorandaconsistentestimatorofthemeansquarederror.Section2.4providessucientconditionsforasymp-LocalSurveyRegressionEstimators2toticnormalityofthelocalpolynomialregressionestimatorandestablishesacentrallimittheoreminthecaseofsimplerandomsampling.WeshowthattheestimatorisrobustinthesenseofasymptoticallyattainingtheGodambe-JoshilowerboundtotheanticipatedvarianceinSection2.5,aresultpreviouslyknownonlyfortheparametriccase.Section3reportsonasimulationstudyofthedesignpropertiesoftheestimator,whichiscompetitivewiththeclassicalsurveyregressionestimatorwhenthepopulationregressionfunctionislin-ear,butdominatestheregressionestimatorwhentheregr