Model Based Estimation of Population Parameters with Applications

Abstract

The problemoffinitepopulationparameterestimation,insuperpopulationsettings,is receivingconsiderableattentioninthefieldofsurveysampling.Inthisdissertation,we developageneralframeworkofmodel-basedapproachforestimationoffinitepopulation parameter t (a linearcombinationofpopulationvalues),assumingsuperpopulationsetting under basisfunctionregressionmodel.Bayesianversionoftheproposedgeneralframework is alsostudiedbyassumingtheGaussiandistributionfortheerrorterm,forincorporating prior informationaboutthesuperpopulationparameters.Specialcasesoftheproposed general frameworkaredeductedtoobserveitsapplicability.Expressionsforpredictionerror varianceandmodel-biasoftheproposedestimator ˆt are derived.Forstatisticalinference about t, estimationofpredictionerrorvarianceunderdifferentmodelselectioncriteria residual, generalizedcrossvalidation(GCV),unbiasedestimatedvariance(UEV),final prediction error(FPE)andBayesianinformationcriteria(BIC)methods,isalsoconsidered. An indexforincreaseinefficiencyonusingadditionalbasisfunctions,namedasincrement in efficiency(IE), isalsodevisedunder,simple,ridgeandBayesianregression.Theindex providesalogicalguidelineforselectingamodelwithappropriatenumberofbasisfunctions or covariates.Non-responseprobleminthestudyvariableisdealtbasedonsub-sampling technique, knownasHansenandHurwitztechnique,undermodel-basedapproachwiththe assumption thattherespondingandnon-respondingpopulationhavedifferentmodelsand the occurrenceofnon-responseisobservablejustlikeastratificationvariableinstratified sampling. Design-basedefficiencycomparisonsaremadebasedonrealandsimulateddata sets. Underlinearpopulationmodel(linearinparameteraswellasinvariables),thetotal estimator withsub-samplingismodel-unbiasedandhassmallermodel-varianceascompared to predictiveestimatorbasedonsampledrespondentsonly.Wepresentsnewversionof rankedsetsamplingforobtainingmoredispersedunitswithtitle,therankedsetsampling without replacement(RSSWOR),basedontheassumptionthatthefinitepopulationis coming fromaninfinitesuperpopulationviasomestochasticprocesswithfinitemeanand variance.BothmathematicalexpressionsandMonte-Carloexperimentsupportthesuperiority of thetotalestimatorunderRSSWORoverthecompetitorsundersimplerandomsampling without replacement(SRSWOR)foraspecialmodel,socalled,gammapopulationmodel (GPM). Estimationofsub-populationtotalunderanewversionofrankedsetsamplingfor obtaining awithoutreplacementsamplewithGPM(generalformofproportionalpopulation model) isalsoprovided.Further,themodelrelationshipbetweenthestudyvariableandthe auxiliary variableforwholepopulationisusedtopredictthenon-sampledvaluestoestablish a domainspecificestimatorfortotal.Thesuperiorityofthedomainspecifictotalestimator under RSSWORoverthetotalestimatorunderSRSWORforspecificcasesarealsoshown mathematically aswellasthroughMonte-Carloexperiment. Finally,weanalyzethebirthhistorydatafromPakistanDemographicHealthSurvey ix (PDHS) 2017-18usingthreeseparatemodelstaking1-yearperiodbirthsforfirst,3-years period birthforsecondand5-yearsperiodbirthsforthirdmodelsastheresponsesand24 regressorsinsteadofbasisfunctionofsingleregressor.Mainly,thePoissonregressionmodel with log-linkfunctionisusedformodelingpurpose.Todealwithresponseshavinglarge varianceandmany0’saswellasafewverylargevalues,weusenegativebinomial(NB), zero inflatedPoisson(ZIP)andzeroinflatednegativebinomial(ZINB)modelsasextensions of Poissonmodel.WealsoconducttheestimationofregressionmodelsunderBayesian paradigm assumingnormalpriorsforeachcoefficientsincludingintercept.Theposterior means areobtainedusingrjags(RJustAnotherGibbsSampler)packageinRStatistical software.Theposteriormeansforeachcoefficientsareobservedclosertoclassicalestimates for Poissonmodels.Somemodeldiagnosticsareappliedtocheckthevalidityofestimation procedure. Themodel-basedfertilityratesi.e.agespecificfertilityrate(ASFR),totalfertility rate (TFR),generalfertilityrate(GFR)andgrossreproductionrate(GRR)areobtainedusing predicted responseundertheestimatedmodels.Weprovideanillustrationofpredictive approach throughbootstrapsamplingfromthePDHS2017-18individualrecodedata.