7、Sampling-with-unequal-probabilities

Chapter7.Samplingwithunequalprobabilities1.Concept:Ifeveryunitinthepopulationhasanunequalprobabilitytobechoseninthesample,wecallitSamplingwithunequalprobabilities.Theprobabilitiesareoftenrelatedwithsomeauxiliaryvariables.suchasunits’SizeMeasures,informationhighlyrelatedwithsurveyvariables7.1Theconceptandreasonsiyi120045022004003600125041503505100180690020007100023008250510915035010250510total38008300ixiyixExample:Thereare10townsinthecity,eachtowncontainsseveralvillages,meansthetotalyieldsofcropsofithtown,meansthetotalplantingareasofithtown.wewanttoknowtheaverageyieldforthetowninthecity(ortotalyieldsforthecity)IfweuseSRS,theneachtownhasthesameprobabilitytobechosen,thatis1/10.It’sunfair!tunmuItiselementsampling1202004502142004003566001250414150350591001806869002000710510002300821250510913150350102425051036238008300xyExample:Thereare10townsinthecity,eachtowncontainsseveralvillages,andthenumberofallvillagesis362,meansthenumberofvillagesthetowncontains,meansthetotalyieldsofcropsforthetown,meansthetotalplantareasforthetown,wewanttoknowthetheaverageyieldforthevillageandtotalyieldsofcropsforthecityBxyBUnitsizeAuxiliaryvariableItisclustersamplingIfthetownsarechosenwithSRS,theneachtownhasthesameprobabilitytobechosen,thatis1/10.It’sunfair!2.TheconditionofusingitSamplingwithunequalprobabilitiesisoftenusedunderfollowingcircumstances:1、Toestimatethepopulationtotal,butthetheunits’sizesaresixtoone(相差悬殊）.2、theunits’sizesaretoolarge,thencannotchoosethebasicrelativelysmallunits.Choosingsamplingunitswithunequalprobabilitiescanimprovetheprecision,buttheconditionisweshouldknowanauxiliaryvariabletomaketheprobabilityforeveryunit.3.Thereason1.withreplacementandwithoutreplacementAccordingtowhethertheunitsreplacetothepopulationornotAstoSamplingwithunequalprobabilities,replacementisoftenused2.PPSandPPZAccordingtoeithertheprobabilitiesareproportionaltotheunits’sizesortothevalueoftheauxiliaryvariable7.2ThecategoryofSamplingwithunequalprobabilitiesPPS:samplingwithprobabilityproportionaltoPS’ssizePPZ:samplingwithprobabilityproportionaltothevalueoftheauxiliaryvariableArebothreplacementinthischapter1202004502142004003566001250414150350591001806869002000710510002300821250510913150350102425051036238008300xyExample:Thereare10townsinthecity,eachtowncontainsseveralvillages,andthenumberofallvillagesis362,meansthenumberofvillagesthetowncontains,meansthetotalyieldsofcropsforthetown,meansthetotalplantareasforthetown,wewanttoknowthetotalyieldsofcropsforthecityBIfweusePPZ,theprobabilityofeachtownchosentothesampleis:1aAaaBBxyBIfweusePPS,theprobabilityofeachtownchosentothesampleis:1aAaxxUnitsizeAuxiliaryvariableInpractice,therearetwowaystooperatingtheSamplingwithunequalprobabilities.A)Hansen-Hurwitzmethod累计总和法7.3TheoperationofSamplingwithunequalprobabilities先把总体各单位按一定顺序排列,并依次列出辅助标志值的累计数，然后在1至累计数总数之间随机抽选一个数字，那么累计数大于等于该随机数的单位即为中选单位。Example:Thereare8streets,wewanttochoosethestreetsaccordingtothenumberofpeopleitholds(10000)Weshouldfixonanumberbyrandomfrom1to45.1,if15,sothestreet3ischosen,because15.1＞15.if25,thestreet6ischosen(therandomnumbersfrom24.3to33.1areallmeanthestreet6ischosen)Themorepeoplethestreetholds,thelargerprobabilityis.So,thehighertheauxiliaryvariableis,thelargertheprobabilityis.streetNumberofpeoplestreetNumberofpeople13.53.553.924.227.611.168.933.134.015.176.139.245.220.385.945.1ixixixixB)LahirismethodTheshortcomingiswhenthepopulationislarge,listingallelementsisdifficult.先找出最大的辅助变量值，设为Max{yi},然后在1至总体单位数N之间随机抽选一个数设为i，编号i单位的辅助变量值为yi，再在1至Max{yi}之间随机确定一个数设为y0，当yi≥y0时，编号i的单位就被抽中。若yi＜y0重复上述过程，直至抽出n个单位。Example:Thereare8streets,wewanttochoosethestreetsaccordingtothenumberofpeopleitholds(10000)Max{yi}=8.9，atfirst,wefixonanumberbyrandomfrom1to8,if5,y5=3.9.then,chooseanumberfrom1to8.9,if3,becauseyi=3.9＞y0=3,thestreet5ischosen.Ify0≥4,thestreet5isnotchosen,weshouldrenewtheiandy0tochooseanothernumber.streetNumberofpeoplestreetNumberofpeople13.53.553.924.227.611.168.933.134.015.176.139.245.220.385.945.1ixixixixyi:thesurveyvariablexi:theauxiliaryvariable:theprobabilityoftheunittobechoseninthesampleletIfthecorrelationcoefficientbetweenxandyispositiveandhigh,thentheestimateofYisgivenasyi:iiipziyyNiiiixx17.4Thebasicprinciplesofsamplingwithunequalprobabilities(PPZ)X:thepopulationtotalofauxiliaryvariableThesamplesizeisn,wecangetnestimatesofYniiiniipzpzynnyy111niiipzpzNynyNy111Thevarianceofandarepzypzy22111()()NiipziiyVarySYnnN22211()()NipziiiyNVarySYnn221111var()()1nipzpziiyysynnnN2222121var()()11()(1)nipzpziiipziyNNysynnnNyynnTheunbiasedestimatorareExample:Thereare33townsinthecity,wechoose10townswithunequalprobabilities,andtheprobabilitiesaremadebasingontheplantingareaofeachtown,thetotalplantingareaofthecityis30525.(someinformationarelistedinthetable,meanstheyieldsofcrops,meanstheplantareas)pleasegivetheinferenceintervalofaverageyieldforthetowns.(a=5%)122800222.8780330.21000421.7700525.3880631.21100726850820.5800933.812001023.6830257.18940iixiyixiyNiiiixx1/12280