[2] y π π ( )π j πj i πi πj i 2. U = {1 K N} y p(s) S i j k I k yij wi = 1 πi πj i I k = 1 k S ^tπ = { i j wiyij 0k S y S πk k πk = Pr(k S)=Pr(I k =1)

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Download "[2] y π π ( )π j πj i πi πj i 2. U = {1 K N} y p(s) S i j k I k yij wi = 1 πi πj i I k = 1 k S ^tπ = { i j wiyij 0k S y S πk k πk = Pr(k S)=Pr(I k =1)"

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1 26 10 Vol.26 No.10 Statistics&InformationForum Oct.2011 ab b ( a. ;b ) ; ; ; C811 A (2011) Neyman Neyman [1] (10JJD790036); (11BTJ009) ; 3

2 [2] y π π ( )π j πj i πi πj i 2. U = {1 K N} y p(s) S i j k I k yij wi = 1 πi πj i I k = 1 k S ^tπ = { i j wiyij 0k S y S πk k πk = Pr(k S)=Pr(I k =1)= p(s) k S T = U yk π V(^tπ )=V( s ) yk)= U Δkl ) yk ) yl (2) 1.π Δkl =πkl-πkπlπkl k l πkl <πkπl ) yk =yk/πk ) yl =yl/πl k l Uπkl >0 V(^tπ ) 2. π ^V(^tπ )=^V( s ) yk)= S ) Δkl ) yk ) yl (3) ) [3]42-45 Δkl = Δkl/πkl 3. π ( ) π ) p(s) i πi ^tπ = U Ik yk yk = (1) S πk πk y yk y k S [4] π ( ) Horvitz- Thompson Y 1/πk yk 1/πk 4 4. ( 1.

3 5. Y = Y i + π i S Y i (5) i S S Cassel S rndal Wret- man 珔 x 1N-n = (N -n) -1 (N 珔 x N - i ) i Sx 珚 [3]42-45 Y N 珚 Y reg = N -1^Y (7) 珚 Y reg AV( 珚 Y reg - 珚 Y N )= E{E[( 珚 Y reg - 珚 Y N ) 2 F N ]}- ( ) 1. y 2. y y 0 σ 2 3. ( ) N ( ) 1. yi 烄 =x iβ+e i 烅烆 e i ~ind(0γiσ 2 )(i=1 N ) yi x i (k+1) x i = (1x i1 x i2 x ik ) β = ( β 0 β 1 β k) e i i je i x i β X [5] ^β = (X D γ -1 X) -1 X D γ -1 y (4) X = (x 1 x 2 x n ) D γ =diag(γ11γ22 γnn)y = (y1y2 yn) n S Y ^Y = i Syi + i S yi = i S yi + (N -n) 珔 x 1N-n^β (6) [E{E[( 珚 Y reg - 珚 Y N ) F N ]}] 2 (8)

4 [12] [6] Sugden Smith [13] Pfefermann [14] Kot [15] ; Sterba 1 [4] Kot Demets Halperin Nathan Holt Hausman Wise Jewel 1. Y Y = Xβ +ε β ^β = [7-8][9] [10] Skinner Holt Smith (X X) X y ^β = (X WX) X Wy W [11] Pfef- 2. fermann Pfef- fermann Skinner Holmes Goldstein Rasbash Kovacevic Rai Rabe - Hesketh Skrondal [16-18] p(s)

5 [1] Hansen M HMadow W GTeppingBJ.AnEvaluationofModel-DependentandProbability-SamplingInferencein SampleSurveys[J].JournaloftheAmericanStatisticalAssociation198378(384). [2]. [J] (1). [3] CasselC MS rndalcewretmanjh.modelassistedsurveysampling[m].new YorkSpringer1992. [4] SterbaSK.AlternativeModel-BasedandDesign-BasedFrameworksforInferencefrom SamplestoPopulations[J]. MultivariateBehaviorResearch200944(6). [5] FulerW.SamplingStatistics[M].New YorkJohn Wiley2009. [6] [M] [7] DemetsDHalperin M.EstimationofaSimpleRegressionCoeficientsinSamplesArisingfromSub-SamplingProce- dures[j].biometrics1977(1). [8] NathanGHoltD.TheEfectofSurveyDesignonRegressionAnalysis[J].JournaloftheRoyalStatisticalSociety1980 (3). [9] HausmanJAWiseD A.StratificationonEndogenousVariablesandEstimationTheGaryIncomeMaintenanceExperi- ment.structuralanalysisofdiscretedatawitheconometricapplications.cambridge[m].massmitpress1981. [10]JewelN P.LeastSquaresRegressionwithDataArisingfromStratifiedSamplesoftheDependentVariable[J].Biometri- ka198572(1). [11]SkinnerCJHoltDSmithT M F.AnalysisofComplexSurveys[M].New YorkWiley1989. [12]PfefermannDKriegerA MRinotY.ParametricDistributionsofComplexSurveyDataunderInformativeProbability Sampling[J].StatisticaSinica1998(8). [13]SugdenR ASmithT M F.IgnorableandInformativeDesignsinSurveySamplingInference[J].Biometrika1984(3). [14]PfefermannD.TheRoleofSampling Weights When ModelingSurveyData[J].InternationalStatisticalReview1993 (2). [15]KotPS.Randomized-Assisted Model-BasedSurveySampling[J].JournalofStatisticalPlanningandInference (1). [16]PfefermannDSkinnerCJHolmesDJGoldstein HRasbashJ.WeightingforUnequalSelectionProbabilitiesin Mul- tilevelmodels[j].journaloftheroyalstatisticalsocietyseriesb199860(1). nicationsinstatistics2003(1). [17]KovacevicM SRaiSN.A Pseudo Maximum LikelihoodApproachto MultilevelModelingofSurveyData[J].Commu- [18]Rabe-HeskethSSkrondalA.MultilevelModelingofComplexSurveyData[J].JournaloftheRoyalStatisticalSocie- tyseriesa (4). 7

6 26 10 Vol.26 No.10 Statistics&InformationForum Oct.2011 ( ) ; ; ; ; C812 O212 A (2011) (The MakingofIndex Num- ber) (Fisher) ; AComparativeStudyofComplexSamplingInferenceSystems JIN Yong-jin ab HEBen-lan b (a.appliedstatisticalscienceresearchcenter; b.schoolofstatisticsrenminuniversityofchinabeijing100872china) AbstractThereareusualytwoinferencesystemsforcomplexsampletraditionalstatisticalinference andmodel-basedstatisticalinference.traditionalsamplingtheorybasedonrandomizationtheorybelieved thatthevaluesofvariablesonpopulationunitsarefixedandtherandomnessembodiesinsampleselection. Itsinferenceforpopulationdependsonsamplingdesign.Estimatorsofthismethodarerobustwhensample sizeislargebutineficiency whensamplesizeissmalandthereare missingdata.anotherdeduction basedon modelthinksthatthefinitepopulationisarandom sampledrawnfrom asuper-population. Inferenceforpopulationdependsonmodelingbutestimatorsofthisinferencesystemarebiasedundernon -ignorablesamplescheme.basedontheanalysisofthecorecontentsofthetwo methodsthispaper proposessamplingscheme-assistedand model-basedinferenceandpointsoutthatthis methodhas importantapplicationvalueincomplexsampling. Keywordsrandomization-based;model-based;samplingscheme;complexsamplinginference ( ) 8