Auxiliary variable

Classification

Intermsoftheirnature,auxiliaryvariablescanbedividedintothefollowingcategories:

1.Informationreflectingtheoverallstructure.Forexample,theoverallsamplingunitcanbedividedintoseverallayersaccordingtothedegreeofdifferenceinthevalueofthesurveyindicators.Thecompositionofeachlayeranditsproportionintheoverallinformationreflecttheoverallstructure.

2.Informationonthescale.Forexample,whenthepopulationisdividedintosamplingunitsofdifferentlevels,toknowthenumberofsamplingunitsofacertainlevel,youcanusethenumberofsamplingunitsofthenextlevelcontainedinitorothermeasurementvalues​​toindicatethesizeorsizeofthissamplingunit.

3.Itistheinformationofauxiliaryindicatorscloselyrelatedtotheinvestigatedindicators.Forexample,wheninvestigatingtheconsumptionexpenditureofresidents,thedisposableincomeofresidentsisanauxiliaryindicatorthathasahighcorrelationwithexpenditure.Theinformationofthisauxiliaryindicatorcanbetheoverall,samplingunitandsample.

4.Relevanthistoricalinformationofsurveyindicators.Forexample,insomeregularsamplesurveys,theprevioussurveyindicatorsareoftenusedasauxiliaryindicatorsforthecurrentsurveyindicators,andsometimestherecentcensusindicatorsofthesamesurveyitemcanbeusedasauxiliaryindicatorsforthecurrentsamplingsurveyindicators.Inthiscase,auxiliaryindicatorsareprovided.Theinformationisthehistoricalinformationofthesurveyindicators.

Role

Theroleofauxiliaryvariablesismainlyreflectedintwoaspects:oneistoimprovethesamplingmethodandincreasetherepresentativenessofthesampletothepopulation.Thesecondistoimprovetheestimationmethod,reducetheestimationerror,andimprovetheestimationaccuracy.Forthefirstaspect,therearemainlythefollowingthreemodesofaction.

1.Theroleofauxiliaryvariablesinstratifiedsampling.

Stratifiedsamplingisasamplingmethodthatisfrequentlyusedinactualwork.Theefficiencyofstratifiedsamplingmainlydependsonthevariancewithintheoverallstrataandthevariancebetweenstrata.Usingauxiliaryvariablestostratifythepopulationcaneffectivelyreducethevariancewithinthelayerandexpandthevariancebetweenthelayers.Reducetheratioofthevariancewithintheoverallstratatothevariancebetweenstrata,therebygreatlyimprovingtheefficiencyofsamplingestimation.

2.Theroleofauxiliaryvariablesinsystematicsampling.

Symmetricandequidistantsamplingissuitableforthesamplingdesignofthelineartrendpopulation,butmostofthepopulationswefaceinactualworkarenon-linear.Usingauxiliaryvariablestoranktheresearchpopulationcanbeusedtosortthenon-linearOveralllineartrending,andimplementingsymmetricandequidistantsamplingonthisbasisisaneffectivewaytoimprovesamplingefficiency.

3.Usetheauxiliaryvariableratioasthesamplingprobabilityofthesamplingunittoimplementunequalprobabilitysampling.

Inparticular,unequalprobabilitysamplingwithclustersasthesamplingunitcaneffectivelyeliminatethe"numberleveleffect"and"ratenumbervariabilityeffect"ofequalprobabilityclustersampling,andimprovetheefficiencyofclustersampling.Usingauxiliaryvariablescanimplementunequalprobabilitysamplingtoimprovesamplingdesign.

Therearealsomorecommonapplicationsforthesecondaspect,forexample,theuseofauxiliaryvariablestoestablishratioestimatorsandregressionestimators.Constructingaratioorregressionestimatorforthepopulationmeanortotalvalueisanimportantmeanstoimprovethesamplingdesignfromtheestimationprocess,butbothoftheseestimationmethodsrequirecorrespondingauxiliaryvariables,andthepopulationmeanortotalvalueoftheauxiliaryvariablesneedstobeobtained.Inaddition,inadditiontotheabovetwoauxiliaryvariables,auxiliaryvariablescansometimesbeusedtodealwithmissingdatainsamplesurveys.

Usingauxiliaryvariablesforsamplingdesigncangreatlyimprovetheaccuracyofestimation.Therefore,atthebeginningofsamplingdesign,weshouldfocusoninvestigatingwhetherthereisauxiliaryvariabledataforselectionanduse,andfurtherconsiderhowtouseitbetter.

Use

Usingauxiliaryvariablesforsamplingdesignhastwopurposes:oneistoimprovethesamplingmethod,andtheotheristoimprovetheestimator.Theninthesamplingsurvey,duetothedifferenceoftheestimateandthesamplingmethod,theauxiliaryvariablemustbeusedaccordingtothesituation.

1.Useauxiliaryvariablestostratifythepopulation

Stratifiedsamplingisasamplingmethodthatisfrequentlyusedinactualwork.Itissuitablefortwoortwoauxiliaryvariables.Morethanonesituation.Theefficiencyofstratifiedsamplingmainlydependsontheratioofthevariancewithintheoverallstratatothevariancebetweenstrata.Usingauxiliaryvariablestostratifythepopulationcaneffectivelyreducetheintra-layervariance,expandtheinter-layervariance,andreducetheratiooftheoverallintra-layervariancetotheinter-layervariance,therebygreatlyimprovingtheefficiencyofsamplingestimation.Thismethodcanbeusedtodeterminetheoptimalstratificationpoint,sothatthevarianceofthetargetquantityestimateisthesmallestunderthesamecircumstances.

2.Useauxiliaryvariablestoqueuetheoverallunits,andthenperformsystematicsampling

Thisisasystematicsamplingofqueuingaccordingtorelevantsigns.Sincetheaccuracyofsystematicsamplingdependsonthevariancewithinthesample,thegreaterthevariancewithinthesample,thesmallerthevarianceofthetargetquantityestimate.Queuingbyauxiliaryvariablescantrendthenonlinearpopulationlinearly.Onthisbasis,theimplementationofsystematicsamplingcanincreasetheintra-samplevarianceoftheobtainedsystem,therebyimprovingtheaccuracy.Theoperationofthismethodissimpleandconvenient,andtheeffectisgenerallyideal.Thedisadvantageisthatitisdifficulttoestimatethevarianceoftheestimator,andbecausethequeuingonlyusestheinformationofthesizeorderoftheauxiliaryvariables,theinformationutilizationisnotverysufficient.

3.Usingauxiliaryvariablesforunequalprobabilitysampling

Unequalprobabilitysampling,especiallytheunequalprobabilitysamplingwiththegroupunitasthesamplingunit,caneffectivelyimprovetheclustersamplingsefficiency.Whenusingauxiliaryvariablesforunequalprobabilitysampling,practicalworkersoftenusethefollowingtwomethods:replacementPPSsamplingandPPSsystematicsampling.Theformersampling,estimatorandvarianceestimationareextremelysimple,buttheaccuracyisslightlyworse;whilethelatterasasystematicsampling,varianceestimationismoredifficult.

4.Theuseofauxiliaryvariablestoestablishratioestimators,andregressionestimatorstoconstructratiosorregressionestimatorstotheoverallmeanortotalvalueareimportantmeanstoimprovesamplingdesignfromtheestimationprocess.

Bothestimationmethodsrequirecorrespondingauxiliaryvariables,andobtaintheoverallmeanortotalvalueoftheauxiliaryvariables.Ratioestimationandregressionestimationareusuallyextremelyeffectivewhentheauxiliaryvariablesarehighlycorrelatedwiththesurveyindicators.Especiallyso.Oneofitsgreatadvantagesisthatitcanbeusedinmulti-indexsituations,inwhichdifferentindexvalues​​oftenusedifferentauxiliaryvariables.Thedisadvantagesoftheestimationorregressionestimationarethatthecalculationismorecomplicated,andtheestimatorisbiased.However,whenthesamplesizeisrelativelylarge,thebiasoftheestimatorisnotlarge.Inthecaseofalargesample,biasaccountsforonlyasmallpartofthetotalmodelerrorrelativetothevariance.

5.Useauxiliaryvariablesforpost-layering.

Sometimesitisdifficulttostratifyinadvance.Withoutastratifiedsamplingframe,stratifiedsamplingcannotbecarriedoutandstratifiedsamplescannotbeobtained.Ifyouwanttousethebenefitsofstratifiedsamplingtoimproveaccuracyandgetanestimateofeachsub-populationatthistime,youmustusethepost-stratificationtechnique.Oneoftheprerequisitesfortheadoptionofexpoststratificationisthatthestratificationrightscanbeobtainedthroughsomemeansandthusareknown.Subsequentstratificationhaslessdemandforauxiliaryinformation.Itdoesnotneedtograsptheinformationofeachunitofthewhole,butonlyneedstounderstandsomekindofsummaryinformation;therefore,thecostislowerandtheeffectispoor.Itisnolongervalid.

Choice

Atthebeginningofthesamplingdesign,weshouldfocusoninvestigatingwhetherthereareauxiliaryvariablesthatcanbeused,andfurtherconsiderwhatvariablescanbeusedasauxiliaryvariables,whichauxiliaryvariablesarebetter,andhowChoosetheappropriateauxiliaryvariablefromalargenumberofvariables.

1.Waystoobtainauxiliaryvariables

1)Historicaldata

Statisticalsurveysgenerallyuseacombinationofperiodiccensusandregularsamplingsurveysmethod.Periodiccensusesprovidealargeamountofhistoricaldataforsamplesurveysduringthetwocensuses.Itincludeshistoricaldataofthesurveyvariablesthemselvesandothervariablesrelatedtothesurveyvariables.Thesedatagenerallyhavelowacquisitioncostsanddataaccuracy.High,ithastheadvantagesofhighcorrelationwithresearchvariables.

2)Currentrelateddata

Someresearchvariableshaverelateddataofthesameperiod.Forexample,policestationsgenerallyhaverelativelycompletedemographicdatawithintheirjurisdiction;administrativedepartmentsforindustryandcommercehaverelativelycompletebusinessregistrationdatawithintheirjurisdiction.Thesedatahavecorrelationswithmanyvariables,andcanbeusedasauxiliaryvariablesfortheresearchvariablesinthesamplingdesign.

3)Pilotsurvey

Ifoveralldataisnotavailable,apilotsurveycanbeconsideredtoobtaintheauxiliaryvariabledatarequiredbythesamplingdesign.Samplesforexperimentalsurveyscanbedrawneitherbyprobabilitysamplingmethodsornon-probabilitysamplingmethods.

4)Doublesampling

Ifthereisnoconvenientauxiliaryvariable,youcanalsoconsiderusingthedoublesamplingtechnique.First,drawarelativelylargesimplerandomsamplefromthepopulationtoestimatetheauxiliaryvariable.Then,takethissampleasasmallpopulation,andcarryoutthesamplingdesignofthenecessaryauxiliaryvariables.Underthecircumstancethatthetotalsurveycostisconstant,takingandsurveyingthefirstre-samplewillofcoursecostapartofthecost,sothesamplesizeofthesecondre-samplehastobereduced.Thecostofthefirstre-samplingistoobtaintheinformationofauxiliaryvariablesinordertoimprovetheestimationaccuracyofthesecondre-sampling;thesecondre-samplingwillalsocausethelossofestimationaccuracyduetohavingtoreducethesamplesize.Comparingthetwo,onlytheformer'sgaininimprovingaccuracyexceedsthelatter'slossinreducingaccuracy,theuseofdoublesamplingtechnologyismeaningful.

2.Theprincipleofselectingauxiliaryvariables:theprincipleofbesteffect

Samplingsurveyshaveaninput-outputproblem.Theinputisthesurveyfunding,andtheoutputistheestimatedvalueoftheoverallindex,Andusethesamplingaccuracytomeasure.Inthisway,thebesteffectprinciplecanbedecomposedintotwoaspects:thehighestsamplingaccuracyprincipleandthelowestsurveycostprinciple.

Theformerrequiresthatwhenselectingauxiliaryvariables,considerationshouldbegiventoadaptingtheselectedauxiliaryvariablestotherequirementsofsamplingandestimationmethodsinordertoachievethehighestsamplingaccuracy.Thisisbecausedifferentsamplingandestimationmethodshavedifferentrequirementsfortherelationshipbetweenauxiliaryvariablesandresearchvariables.Forexample,theauxiliaryvariablesusedforstratificationandregressionestimationshouldhaveahighlinearcorrelationwiththeresearchvariables.;TheauxiliaryvariablesusedforPPSsamplingandratioestimationshouldhaveaproportionalrelationshipwiththeresearchvariables.Thelatterrequiresthatwhenselectingauxiliaryvariables,considerationshouldbegiventothelowestcostunderagivensamplingaccuracy.Thisisbecausethecostofobtainingthevalueofdifferentauxiliaryvariablesisverydifferent.Someauxiliaryvariablesmaybeobtainedatarelativelylowcost,whilesomeauxiliaryvariablevalues​​maybeobtainedataconsiderablecost.Auxiliaryvariablesarevariablesintroducedtoimprovetheaccuracyofestimation.Underthecircumstanceofcertainsurveyexpenses,themoreexpensesoccupiedintheinvestigationofauxiliaryvariables,thelesssurveyexpensesleftforthesample,whichwillaffectthesample.capacity.

Auxiliaryvariablemethod

(auxiliaryvariablemethod)

Auxiliaryvariablemethodisalsocalledinstrumentalvariablemethod.Animprovedleastsquaresparameterestimationmethod.

Suppose

wherey(t)isthemeasurement,φ(t)istheregressionvector,θ₀Istheparametervectortobeestimated(Tstandsfortransposition),v(t)isthecorrelatednoise,thentheleastsquareestimation

doesnotconvergetothetruevalueθ.Inthiscase,anauxiliaryvectorz(t)canbeused,aslongas:

1,z(t)andv(t)arenotrelatedtoeachother,thatis,

2.Thematrixisinvertible.

3,themeanofv(t)iszero.

Itcanbeprovedthatwhen,theauxiliaryvariableestimator

willtendto(ieconvergeand)thetruevalueθ₀.

Thekeytotheauxiliaryvariablemethodishowtoconstructanauxiliaryvariablethatsatisfiestheaboveconditions,thatistosay,theauxiliaryvariableshouldnotbecorrelatedwithnoisev(t),butstronglycorrelatedwithφ(t).Therefore,acommonchoiceistohaveinputtostimulateadeterministicsystemtoproduceoutputandinputcomponents,andtheotheristotakealldelayedinputsasauxiliaryvariables.Thebasicideaof​​theauxiliaryvariablemethodisveryvaluable.Appropriateselectionofauxiliaryvariablescancommunicatewithotheridentificationmethods.Therefore,itcanbeusedinmanyoccasions.Ofcourse,thereisalsotherecursiveauxiliaryvariablemethod.Itisalsowidelyused.use.