Some contributions to small area estimation

2004·5.3 MB·English

by  Sinha, Karabi

SOMECONTRIBUTIONSTO
SMALLAREAESTIMATION
By
KARABISINHA
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2004
Copyright2004
by
KarabiSinha
ToBabaandMaa.
ACKNOWLEDGMENTS
Mysincerestgratitudeistomyadvisor,ProfessorMalayGhosh,forhis
unyieldingsupportandinvaluableguidancethroughoutmygraduatework. Noneof
thiswouldhavebeenpossiblewithouthim.
MysincerethanksareduetoProfessorDalhoKimfortheimmensehelpI
receivedfromhimincomputation. Iwouldalsoliketothankallmycommittee
members.ProfessorsRonaldRandles,GeorgeCasella,GyndiGarvanandBruce
Vogelfortheirconstructivecommentsregardingmydissertation.
Therearesomewithoutwhomthisjourneywouldnothavebeenpossible.
Frommyundergraduatedays,Ioweeverythingtomyteacher.ProfessorKashinath
Chatterjee. HisloveforStatistics,forhisstudents,hisartofteaching(yes,he
makesitanart)andhisdevotioninspiredme(andstillcontinuetodoso).
AttheUniversityofFlorida,IhavebeenveryfortunatetohavehadGyndi
GarvanasmymentorinBiostatistics. Shehasbeenextremelykind,encouraging
andinspiring. Myexperiencesasastatisticianindifferentreal-lifeapplicationsand
projectshaveprovedtobeinvaluableandIoweitalltoher.
Finally,therearesomewithoutwhommylifewouldnothavebeencomplete.
Dola-mashinotonlymadethisa“homeawayfromhome”butshebecamemybest
friend. Sheisapillarofstrengthandtouchesthelivesandheartsofallthatcome
incontactwithher. SheisatruefriendandIfeelhonoredandgratefultohaveher
inmylife.
Mydearfriends.Carmen,ToniandBelkys,myfellowpartnersinthetoilsand
joysofgraduateschoollife,arelikemysistersandIconsidermyselffortunateto
havetheminmylife.
IV
Mylittlebrother,one-and-a-halfyearsyoungertome,ismybuddy,my
partner,myjoyandmyconstantsourceofinspiration.
Withoutmymother,Idon’tthinkIwouldhavemadeit. Ifevercaughtin
agloomymomentoramomentofdoubt,shewouldbethepersontotalkto.
Maaurges,encourages,inspiresandwillnothear,foramoment,anythinginthe
negative. SheisthemostpositivepersonIknowandseesmethroughmygloomy
daysandtakesprideinmybest.
Finally,totheonewhochampionsmethrougheverything,totheoneIlookup
toforeverything,tomyfather,toBaba,Ioweeverything.
V
TABLEOFCONTENTS
pase
ACKNOWLEDGMENTS
iv
LISTOFTABLES viii
LISTOFFIGURES x
ABSTRACT
xi
CHAPTER
1 SRMEAVLILEWAREAESTIMATION:ANOVERVIEWANDASELECTIVE
1
1.1 Introduction 1
1.2 SAETechniques: ABriefDescription 3
1.3 SomeIllustrativeExamples 5
1.4 SomeMajorReferences 5
1.5 UseofEBandHBEstimatorsinSAE 6
1.6 DiscussionoftheWorkofFayandHerriot 14
1.7 ConsiderationsfortheCalculationofBayesRiskoftheEBesti-
mator 17
1.8 HBEstimatorsandCorrespondingBayesRisks 18
1.9 Normality-basedPrediction 20
1.9.1 PredictioninMultivariateNormalPopulation 20
1.9.2 BayesPredictionofDomainMeansinFinitePopulation
Sampling 21
1.9.3 EBPredictionofDomainMeansinFinitePopulationSam-
pling 22
1.10 UseofVarianceComponentsModelsinSAE 24
1.10.1 PredictioninVarianceComponentsModel 25
1.10.2Estimationofcr^ 26
1.10.3Estimationofcr^ 26
1.10.4PredictiononUj 27
1.10.5Predictionon7j 27
1.11 MeasurementErrorModels 29
1.12 DiscussionoftheWorkofGhoshetal(1998) 30
1.13 SalientFeaturesandLayoutofthisDissertation 31
VI
2 EMPIRICALBAYESESTIMATIONINFINITEPOPULATIONSAM-
PLING
UNDERFUNCTIONALMEASUREMENTERRORMODELS 33
2.1 Introduction 33
2.2 EBEstimators 35
2.3 BayesRisks 38
2.4 SimulationStudy 41
3 EMPIRICALANDHIERARCHICALBAYESESTIMATION
INFINITEPOPULATIONSAMPLING
UNDERSTRUCTURALMEASUREMENTERRORMODELS 49
3.1 Introduction 49
3.2 EBEstimators 50
3.3 AsymptoticOptimalityoftheEBPredictor 55
3.4 HBPredictors 56
3.5 SimulationStudy 60
3.6 DataAnalysis 61
4 EMPIRICALANDHIERARCHICALBAYESESTIMATIONFORBI-
NARYRESPONSE 69
4.1 Introduction 69
4.2 HBModel 70
4.3 EBestimation 72
4.4 Dataanalysis 83
4.4.1 Selectionofcovariates 84
4.4.2 SmallDomainEstimatesforAsians 85
5 CONCLUDINGREMARKSANDSCOPEFORFUTUREWORK ... 104
APPENDIX 105
REFERENCES 126
BIOGRAPHICALSKETCH 129
vii
LISTOFTABLES
Table page
2-1 Thesamplesizes(nj),thepopulation(“true”)means(TM),thesam-
plemeans(SM),theregressionestimates(R),theempiricalBayeses-
timates(EB)andthecorrespondingRMSE’sforthe12stratawhen
2- 0-^=0 43
32--2 Thepopulationmeans(TM),thesamplemeans(SM),theregression
estimates(R),theempiricalBayesestimates(EB),andthecorrespond-
ingRMSE’sforthe12stratawhencr^=2.5 44
2-3 Thepopulationmeans(TM),thesamplemeans(SM),theregression
estimates(R),theempiricalBayesestimates(EB),andthecorrespond-
3- ingRMSE’sforthe12stratawhencr^=4 45
4-4 Tcohuentpioepsuwlahteinonbime=an5,anthde3e=st4i.m5atorsandtheirRMSE’sforthe12 46
1 Thesamplesizes,meansandRMSE’sforthe12counties 61
3-2 SurveyandSatelliteDataforSoybeansin12Iowacounties 62
3-3 PredictedHectaresofSoybeanWithStandardErrorsofAlternative
Predictors 63
3-4 PredictedHectaresofSoybeanWithStandardErrorsofAlternative
Predictors,usingsoybeanpixelsastheonlycovariate 65
5 PredictedHectaresofSoybean(HBandBHF)WithCorresponding
StandardErrors,usingsoybeanpixelsastheonlycovariate 66
1 DefinitionofDomainsforAsians 89
4-2 DefinitionofDomainsforAsians 90
4-3 DefinitionofDomainsforAsians 91
4-4 SmallAreaEstimatesoftheProportionsofUninsuredAsian: year1997 92
4-5 SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
1997(continued) 93
4-6 SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
1997(continued) 94
viii
4-7 SmallAreaEstimatesoftheProportionsofUninsuredAsian: year1998 95
1998
4-8 SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
(continued) 96
4-9 SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
1998(continued) 97
4-10SmallAreaEstimatesoftheProportionsofUninsuredAsian: year1999 98
4-11SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
1999(continued) 99
4-12SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
21090909(continued) 100
4-13SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
22000000 101
4-14SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
(continued) 102
4-15SmallAreaEstimatesoftheProportionsofUninsuredAsian: year
(continued) 103
IX
LISTOFFIGURES
Figure page
2-1 47
2-2 48
X