Small Area Estimation

451 Pages·2015·3.71 MB·English

by  J.N.K. Rao

SMALL AREA
ESTIMATION
Second Edition
J.N.K.RAOANDISABELMOLINA
WileySeriesinSurveyMethodology
Copyright©2015byJohnWiley& Sons,Inc.
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey
PublishedsimultaneouslyinCanada
LibraryofCongressCataloging-in-PublicationData:
Rao,J.N.K.,1937-author.
Smallareaestimation/J.N.K.RaoandIsabelMolina.–Secondedition.
pagescm–(Wileyseriesinsurveymethodology)
Includesbibliographicalreferencesandindex.
ISBN978-1-118-73578-7(cloth)
1.Smallareastatistics.2.Sampling(Statistics)3.Estimationtheory.I.Molina,Isabel,1975-author.
II.Title.III.Series:Wileyseriesinsurveymethodology.
QA276.6.R3442015
519.5′2–dc23
2015012610
PrintedintheUnitedStatesofAmerica
CONTENTS
ListofFigures xv
ListofTables xvii
ForewordtotheFirstEdition xix
PrefacetotheSecondEdition xxiii
PrefacetotheFirstEdition xxvii
1 *Introduction 1
1.1 WhatisaSmallArea? 1
1.2 DemandforSmallAreaStatistics, 3
1.3 TraditionalIndirectEstimators, 4
1.4 SmallAreaModels, 4
1.5 Model-BasedEstimation, 5
1.6 SomeExamples, 6
1.6.1 Health, 6
1.6.2 Agriculture, 7
1.6.3 IncomeforSmallPlaces, 8
1.6.4 PovertyCounts, 8
1.6.5 MedianIncomeofFour-PersonFamilies, 8
1.6.6 PovertyMapping, 8
2 DirectDomainEstimation 9
2.1 Introduction, 9
2.2 Design-BasedApproach, 10
2.3 EstimationofTotals, 11
2.3.1 Design-UnbiasedEstimator, 11
2.3.2 GeneralizedRegressionEstimator, 13
2.4 DomainEstimation, 16
2.4.1 CaseofNoAuxiliaryInformation, 16
2.4.2 GREGDomainEstimation, 17
2.4.3 Domain-SpecificAuxiliaryInformation, 18
2.5 ModifiedGREGEstimator, 21
2.6 DesignIssues, 23
2.6.1 MinimizationofClustering, 24
2.6.2 Stratification, 24
2.6.3 SampleAllocation, 24
2.6.4 IntegrationofSurveys, 25
2.6.5 Dual-FrameSurveys, 25
2.6.6 RepeatedSurveys, 26
2.7 *OptimalSampleAllocationforPlannedDomains, 26
2.7.1 Case(i), 26
2.7.2 Case(ii), 29
2.7.3 Two-WayStratification:BalancedSampling, 31
2.8 Proofs, 32
2.8.1 ProofofŶ (𝐱)=𝐗, 32
GR
2.8.2 DerivationofCalibrationWeights𝑤∗, 32
j
2.8.3 ProofofŶ =𝐗̂T𝐁̂ whenc =𝝂T𝐗, 32
j j
3 IndirectDomainEstimation 35
3.1 Introduction, 35
3.2 SyntheticEstimation, 36
3.2.1 NoAuxiliaryInformation, 36
3.2.2 *AreaLevelAuxiliaryInformation, 36
3.2.3 *UnitLevelAuxiliaryInformation, 37
3.2.4 Regression-AdjustedSyntheticEstimator, 42
3.2.5 EstimationofMSE, 43
3.2.6 StructurePreservingEstimation, 45
3.2.7 *GeneralizedSPREE, 49
3.2.8 *Weight-SharingMethods, 53
3.3 CompositeEstimation, 57
3.3.1 OptimalEstimator, 57
3.3.2 Sample-Size-DependentEstimators, 59
3.4 James–SteinMethod, 63
3.4.1 CommonWeight, 63
3.4.2 EqualVariances𝜓 =𝜓, 64
i
3.4.3 EstimationofComponentMSE, 68
3.4.4 UnequalVariances𝜓, 70
i
3.4.5 Extensions, 71
3.5 Proofs, 71
4 SmallAreaModels 75
4.1 Introduction, 75
4.2 BasicAreaLevelModel, 76
4.3 BasicUnitLevelModel, 78
4.4 Extensions:AreaLevelModels, 81
4.4.1 MultivariateFay–HerriotModel, 81
4.4.2 ModelwithCorrelatedSamplingErrors, 82
4.4.3 TimeSeriesandCross-SectionalModels, 83
4.4.4 *SpatialModels, 86
4.4.5 Two-FoldSubareaLevelModels, 88
4.5 Extensions:UnitLevelModels, 88
4.5.1 MultivariateNestedErrorRegressionModel, 88
4.5.2 Two-FoldNestedErrorRegressionModel, 89
4.5.3 Two-LevelModel, 90
4.5.4 GeneralLinearMixedModel, 91
4.6 GeneralizedLinearMixedModels, 92
4.6.1 LogisticMixedModels, 92
4.6.2 *ModelsforMultinomialCounts, 93
4.6.3 ModelsforMortalityandDiseaseRates, 93
4.6.4 NaturalExponentialFamilyModels, 94
4.6.5 *Semi-parametricMixedModels, 95
5 EmpiricalBestLinearUnbiasedPrediction(EBLUP):Theory 97
5.1 Introduction, 97
5.2 GeneralLinearMixedModel, 98
5.2.1 BLUPEstimator, 98
5.2.2 MSEofBLUP, 100
5.2.3 EBLUPEstimator, 101
5.2.4 MLandREMLEstimators, 102
5.2.5 MSEofEBLUP, 105
5.2.6 EstimationofMSEofEBLUP, 106
5.3 BlockDiagonalCovarianceStructure, 108
5.3.1 EBLUPEstimator, 108
5.3.2 EstimationofMSE, 109
5.3.3 ExtensiontoMultidimensionalAreaParameters, 110
5.4 *ModelIdentificationandChecking, 111
5.4.1 VariableSelection, 111
5.4.2 ModelDiagnostics, 114
5.5 *Software, 118
5.6 Proofs, 119
5.6.1 DerivationofBLUP, 119
5.6.2 EquivalenceofBLUPandBestPredictor
E(𝐦T𝐯|𝐀T𝐲), 120
5.6.3 DerivationofMSEDecomposition(5.2.29), 121
6 EmpiricalBestLinearUnbiasedPrediction(EBLUP):
BasicAreaLevelModel 123
6.1 EBLUPEstimation, 123
6.1.1 BLUPEstimator, 124
6.1.2 Estimationof𝜎2, 126
𝑣
6.1.3 RelativeEfficiencyofEstimatorsof𝜎2, 128
𝑣
6.1.4 *Applications, 129
6.2 MSEEstimation, 136
6.2.1 UnconditionalMSEofEBLUP, 136
6.2.2 MSEforNonsampledAreas, 139
6.2.3 *MSEEstimationforSmallAreaMeans, 140
6.2.4 *BootstrapMSEEstimation, 141
6.2.5 *MSEofaWeightedEstimator, 143
6.2.6 MeanCrossProductErrorofTwoEstimators, 144
6.2.7 *ConditionalMSE, 144
6.3 *RobustEstimationinthePresenceofOutliers, 146
6.4 *PracticalIssues, 148
6.4.1 UnknownSamplingErrorVariances, 148
6.4.2 StrictlyPositiveEstimatorsof𝜎2, 151
𝑣
6.4.3 PreliminaryTestEstimation, 154
6.4.4 CovariatesSubjecttoSamplingErrors, 156
6.4.5 BigDataCovariates, 159
6.4.6 BenchmarkingMethods, 159
6.4.7 MisspecifiedLinkingModel, 165
6.5 *Software, 169
7 BasicUnitLevelModel 173
7.1 EBLUPEstimation, 173
7.1.1 BLUPEstimator, 174
7.1.2 Estimationof𝜎2and𝜎2, 177
𝑣 e
7.1.3 *NonnegligibleSamplingFractions, 178
7.2 MSEEstimation, 179
7.2.1 UnconditionalMSEofEBLUP, 179
7.2.2 UnconditionalMSEEstimators, 181
7.2.3 *MSEEstimation:NonnegligibleSampling
Fractions, 182
7.2.4 *BootstrapMSEEstimation, 183
7.3 *Applications, 186
7.4 *OutlierRobustEBLUPEstimation, 193
7.4.1 EstimationofAreaMeans, 193
7.4.2 MSEEstimation, 198
7.4.3 SimulationResults, 199
7.5 *M-QuantileRegression, 200
7.6 *PracticalIssues, 205
7.6.1 UnknownHeteroscedasticErrorVariances, 205
7.6.2 Pseudo-EBLUPEstimation, 206
7.6.3 InformativeSampling, 211
7.6.4 MeasurementErrorinArea-LevelCovariate, 216
7.6.5 ModelMisspecification, 218
7.6.6 Semi-parametricNestedErrorModel:EBLUP, 220
7.6.7 Semi-parametricNestedErrorModel:REBLUP, 224
7.7 *Software, 227
7.8 *Proofs, 231
7.8.1 Derivationof(7.6.17), 231
7.8.2 Proofof(7.6.20), 232
8 EBLUP:Extensions 235
8.1 *MultivariateFay–HerriotModel, 235
8.2 CorrelatedSamplingErrors, 237
8.3 TimeSeriesandCross-SectionalModels, 240
8.3.1 *Rao–YuModel, 240
8.3.2 State-SpaceModels, 243
8.4 *SpatialModels, 248
8.5 *Two-FoldSubareaLevelModels, 251
8.6 *MultivariateNestedErrorRegressionModel, 253
8.7 Two-FoldNestedErrorRegressionModel, 254
8.8 *Two-LevelModel, 259
8.9 *ModelsforMultinomialCounts, 261
8.10 *EBLUPforVectorsofAreaProportions, 262
8.11 *Software, 264
9 EmpiricalBayes(EB)Method 269
9.1 Introduction, 269
9.2 BasicAreaLevelModel, 270
9.2.1 EBEstimator, 271
9.2.2 MSEEstimation, 273
9.2.3 ApproximationtoPosteriorVariance, 275
9.2.4 *EBConfidenceIntervals, 281
9.3 LinearMixedModels, 287
9.3.1 EBEstimationof𝜇 =𝐥T𝜷+𝐦T𝐯, 287
i i i i
9.3.2 MSEEstimation, 288
9.3.3 ApproximationstothePosteriorVariance, 288
9.4 *EBEstimationofGeneralFinitePopulationParameters, 289
9.4.1 BPEstimatorUnderaFinitePopulation, 290
9.4.2 EBEstimationUndertheBasicUnitLevelModel, 290
9.4.3 FGTPovertyMeasures, 293
9.4.4 ParametricBootstrapforMSEEstimation, 294
9.4.5 ELLEstimation, 295
9.4.6 SimulationExperiments, 296
9.5 BinaryData, 298
9.5.1 *CaseofNoCovariates, 299
9.5.2 ModelswithCovariates, 304
9.6 DiseaseMapping, 308
9.6.1 Poisson–GammaModel, 309
9.6.2 Log-NormalModels, 310
9.6.3 Extensions, 312
9.7 *Design-WeightedEBEstimation:ExponentialFamily
Models, 313
9.8 Triple-GoalEstimation, 315
9.8.1 ConstrainedEB, 316
9.8.2 Histogram, 318
9.8.3 Ranks, 318
9.9 EmpiricalLinearBayes, 319
9.9.1 LBEstimation, 319
9.9.2 PosteriorLinearity, 322
9.10 ConstrainedLB, 324
9.11 *Software, 325
9.12 Proofs, 330
9.12.1 Proofof(9.2.11), 330
9.12.2 Proofof(9.2.30), 330
9.12.3 Proofof(9.8.6), 331
9.12.4 Proofof(9.9.1), 331
10 HierarchicalBayes(HB)Method 333
10.1 Introduction, 333
10.2 MCMCMethods, 335
10.2.1 MarkovChain, 335
10.2.2 GibbsSampler, 336
10.2.3 M–HWithinGibbs, 336
10.2.4 PosteriorQuantities, 337
10.2.5 PracticalIssues, 339
10.2.6 ModelDetermination, 342
10.3 BasicAreaLevelModel, 347
10.3.1 Known𝜎2, 347
𝑣
10.3.2 *Unknown𝜎2:NumericalIntegration, 348
𝑣
10.3.3 Unknown𝜎2:GibbsSampling, 351
𝑣
10.3.4 *UnknownSamplingVariances𝜓, 354
i
10.3.5 *SpatialModel, 355
10.4 *UnmatchedSamplingandLinkingAreaLevelModels, 356
10.5 BasicUnitLevelModel, 362
10.5.1 Known𝜎2 and𝜎2, 362
𝑣 e
10.5.2 Unknown𝜎2and𝜎2:NumericalIntegration, 363
𝑣 e
10.5.3 Unknown𝜎2and𝜎2:GibbsSampling, 364
𝑣 e
10.5.4 Pseudo-HBEstimation, 365
10.6 GeneralANOVAModel, 368
10.7 *HBEstimationofGeneralFinitePopulationParameters, 369
10.7.1 HBEstimatorunderaFinitePopulation, 370
10.7.2 ReparameterizedBasicUnitLevelModel, 370
10.7.3 HBEstimatorofaGeneralAreaParameter, 372
10.8 Two-LevelModels, 374
10.9 TimeSeriesandCross-SectionalModels, 377
10.10 MultivariateModels, 381
10.10.1 AreaLevelModel, 381
10.10.2 UnitLevelModel, 382
10.11 DiseaseMappingModels, 383
10.11.1 Poisson-GammaModel, 383
10.11.2 Log-NormalModel, 384
10.11.3 Two-LevelModels, 386
10.12 *Two-PartNestedErrorModel, 388
10.13 BinaryData, 389
10.13.1 Beta-BinomialModel, 389
10.13.2 Logit-NormalModel, 390
10.13.3 LogisticLinearMixedModels, 393
10.14 *MissingBinaryData, 397
10.15 NaturalExponentialFamilyModels, 398
10.16 ConstrainedHB, 399
10.17 *ApproximateHBInferenceandDataCloning, 400
10.18 Proofs, 402
10.18.1 Proofof(10.2.26), 402
10.18.2 Proofof(10.2.32), 402
10.18.3 Proofof(10.3.13)–(10.3.15), 402
References 405
AuthorIndex 431
SubjectIndex 437
FIGURES
3.1 Direct, Census, composite SPREE, and GLSM estimates of Row
Profiles 𝜽M =(𝜃M,…,𝜃M)T for Canadian Provinces Newfoundland
i i1 iA
andLabrador(a)andQuebec(b),forTwo-digitOccupationclassA1. 53
3.2 Direct, Census, composite SPREE, and GLSM estimates of row
profiles𝜽M =(𝜃M,…,𝜃M)T forCanadianprovincesNewfoundlandand
i i1 iA
Labrador(a)andNovaScotia(b),forTwo-digitoccupationclassB5. 54
6.1 EBLUP and Direct Area Estimates of Average Expenditure on Fresh
MilkforEachSmallArea(a).CVsofEBLUPandDirectEstimatorsfor
EachSmallArea(b).AreasareSortedbyDecreasingSampleSize. 171
7.1 Leveragemeasuress versusscaledsquaredresiduals. 231
ii
8.1 Naive Nonparametric Bootstrap MSE Estimates Against Analytical
MSE Estimates (a). Bias-corrected Nonparametric Bootstrap MSE
EstimatesAgainstAnalyticalMSEEstimates(b). 267
8.2 EBLUPEstimates,BasedontheSpatialFHModelwithSARRandom
Effects,andDirectEstimatesofMeanSurfaceAreaUsedforProduction
ofGrapesforEachMunicipality(a).CVsofEBLUPEstimatesandof
DirectEstimatesforEachMunicipality(b).MunicipalitiesareSortedby
IncreasingCVsofDirectEstimates. 268
9.1 Bias (a) and MSE (b) over Simulated Populations of EB, Direct, and
ELLEstimatesofPercentPovertyGap100F forEachAreai. 297
1i
9.2 True MSEs of EB Estimators of Percent Poverty Gap and Average of
BootstrapMSEEstimatorsObtainedwithB=500forEachAreai. 298