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數(shù)學(xué)

線性模型

作者：C.R.Rao/等

出版社：世界圖書出版公司出版時(shí)間：1998-08-01

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目錄

線性模型版權(quán)信息

ISBN：7506238187
條形碼：9787506238182 ; 978-7-5062-3818-2
裝幀：簡(jiǎn)裝本
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所屬分類：
自然科學(xué)
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數(shù)學(xué)

線性模型內(nèi)容簡(jiǎn)介

片斷：
Chapter5isdevotedtoestimationunderexactorstochasticlinearre-
strictions.ThecomparisonoftwobiasedestimatorsaccordingtotheMDE
criterionisbasedonrecenttheoremsofmatrixtheory.Theresultsarethe
outcomeofintensiveinternationalresearchoverthelasttenyearsandap-
pearhereforthefirsttimeinacoherentform.Thisconcernstheconcept
oftheweakr-unbiasednessaswell.
Chapter6containsthetheoryoftheoptimallinearpredictionand
gives,inadditiontoknownresults,aninsightintorecentstudiesabout
theMDEmatrixcomparisonofoptimalandclassicalpredictionsaccording
toalternativesuperioritycriteria.
Chapter7presentsideasandproceduresforstudyingtheeffectofsingle
datarowsontheestimationof.Here,differentmeasuresforrevealing
outliersorinfluentialpoints.includinggraphicalmethods,areincorporated.
Someexamplesillustratethis.
Chapter8dealswithmissingdatainthedesignmatrixX.Afterintroduc-
ingthegeneralproblemsanddefiningthevariousmissingdatamechanisms
accordingtoRubin,wedemonstrate''adjustmentbyfollow-upinterviews"
forlong-termstudieswithdropout.Fortheregressionmodelthemethodof
imputationisdescribed,inadditiontotheanalysisofthelossofefficiency
incaseofareductiontothecompletelyobservedsubmodel.Themethod
ofweightedmixedestimatesispresentedforthefirsttimeinatextbook
onlinearmodels.
Chapter9containsrecentcontributionstorobuststatisticalinference
basedonM-estimation.
Chapter10describesthemodelextensionsforcategoricalresponseand
explanatoryvariables.Here.thebinaryresponseandtheloglinearmodelare
ofspecialinterest.Themodelchoiceisdemonstratedbymeansofexamples.
Categoricalregressionisintegratedintothetheoryofgeneralizedlinear
models.
Anindependentchapter(AppendixA)onmatrixalgebrasummarizes
standardtheorems(includingproofs)thatareofinterestforthebookit-
self,butalsoforlinearstatisticsingeneral.Ofspecialinterestarethe
theoremsaboutdecompositionofmatrices(A.30-A.34),definitematrices
(A.35-A.59),thegeneralizedinverse,andespeciallyaboutthedefiniteness
ofdifferencesbetweenmatrices(TheoremA.7l;cf.A.74-A.78).
Thebookoffersanup-to-dateandcomprehensiveaccountofthetheory
andapplicationsoflinearmodels.
TablesfortheX-and.F-distributionsareprovidedinAppendixB.
2
LinearModels
2.1RegressionModelsinEconometrics
Themethodologyofregressionanalysis,oneoftheclassicaltechniquesof
mathematicalstatistics,isanessentialpartofthemoderneconometric
theory.
Econometricscombineselementsofeconomics,mathematicaleconomics,
andmathematicalstatistics.Thestatisticalmethodsusedineconometrics
areorientedtowardspecificeconometricproblemsandhencearehighly
specialized.Ineconomiclawsstochasticvariablesplayadistinctiverole.
Henceeconometricmodels,adaptedtotheeconomicreality,havetobe
builtonappropriatehypothesesaboutdistributionpropertiesoftheran-
domvariables.Thespecificationofsuchhypothesesisoneofthemaintasks
ofeconometricmodelling.Forthemodellingofaneconomic(orascientific)
relation,weassumethatthisrelationhasarelativeconstancyoverasuffi-
cientlylongperiodoftime(thatis,overasufficientlengthofobservation
period),sinceotherwiseitsgeneralvaliditywouldnotbeascertainable.
Wedistinguishbetweentwocharacteristicsofastructuralrelationship,the
variablesandtheparameters.Thevariables,whichwewillclassifylateron,
arethosecharacteristicswhosevaluesintheobservationperiodcanvary.
Thosecharacteristicsthatdonotvarycanberegardedasthestructureof
therelation.Thestructureconsistsofthefunctionalformoftherelation,
includingtherelationbetweenthemainvariables,thetypeofprobabil-
itydistributionoftherandomvariables,andtheparametersofthemodei
equations.

線性模型目錄

Contents
Preface
Introduction
LinearModels
2.1RegressionModelsinEconometrics
2.2EconometricModels
2.3TheReducedForm
2.4TheMultivariateRegressionModel
2.5TheClassicalMultivariateLinearRegressionModel
2.6TheGeneralizedLinearRegressionModel
TheLinearRegressionModel
3.1TheLinearModel
3.2ThePrincipleofOrdinaryLeastSquares(OLS)
3.3GeometricPropertiesofOLS
3.4BestLinearUnbiasedEstimation
3.4.1BasicTheorems
3.4.2LinearEstimators
3.4.3MeanDispersionError
3.5Estimation(Prediction)oftheErrorTermeand
3.6ClassicalRegressionunderNormalErrors
3.7TestingLinearHypotheses
3.8AnalysisofVarianceandGoodnessofFit
3.8.1BivariateRegression
3.8.2MultipleRegression
3.8.3AComplexExample
3.8.4GraphicalPresentation
3.9TheCanonicalForm
3.10MethodsforDealingwithMuiticollinearity
3.10.1PrincipalComponentsRegression
3.10.2RidgeEstimation
3.10.3ShrinkageEstimates
3.10.4PartialLeastSquares
3.11ProjectionPursuitRegression
3.12TotalLeastSquares
3.13MinimaxEstimation
3.13.1InequaiityRestrictions
3.13.2TheMinimaxPrinciple
3.14CensoredRegression
3.14.1Introduction
3.14.2LADEstimatorsandAsymptoticNormality
3.14.3TestsofLinearHypotheses
TheGeneralizedLinearRegressionModel
4.1OptimalLinearEstimationof
4.2TheAitkenEstimator
4.3MisspecificationoftheDispersionMatrix
4.4HeteroscedasticityandAutoregression
ExactandStochasticLinearRestrictions
5.1UseofPriorInformation
5.2TheRestrictedLeast-SquaresEstimator
5.3StepwiseInclusionofExactLinearRestrictions
5.4BiasedLinearRestrictionsandMDEComparisonwiththe
OLSE
5.5MDEMatrixComparisonsofTwoBiasedEstimators
5.6MDEMatrixComparisonofTwoLinearBiasedEstimators
5.7MDEComparisonofTwo(Biased)RestrictedEstimators
5.7.1SpecialCase:StepwiseBiasedRestrictions
5.8StochasticLinearRestrictions
5.8.1MixedEstimator
5.8.2AssumptionsabouttheDispersionMatrix
5.8.3BiasedStochasticRestrictions
5.9WeakenedLinearRestrictions
5.9.1Weakly(R,r)-Unbiasedness
5.9.2OptimalWeakly(R.r)-UnbiasedEstimators
5.9.3FeasibleEstimators--OptimalSubstitutionofin
5.9.4RLSEInsteadoftheMixedEstimator
PredictionProblemsintheGeneralizedRegressionModel.
6.1Introduction
6.2SomeSimpleLinearModels1
6.3ThePredictionModel
6.4OptimalHeterogeneousPrediction
6.5OptimalHomogeneousPrediction
6.6MDEMatrixComparisonsbetweenOptimalandClassical
Predictors
6.6.1ComparisonofClassicalandOptimalPredictionwith
Respecttothey.-Superiority
6.6.2ComparisonofClassicalandOptimalPredictorswith
RespecttotheX.-Superiorityl
6.7PredictionRegionsl
SensitivityAnalysis
7.1Introduction
7.2PredictionMatrix
7.3TheEffectofaSingleObservationontheEstimationofPa-
rameters
7.3.1MeasuresBasedonResiduals
7.3.2AlgebraicConsequencesofOmittinganObservation
7.3.3DetectionofOutliers
7.4DiagnosticPlotsforTestingtheModelAssumptions
7.5MeasuresBasedontheConfidenceEllipsoid
7.6PartialRegressionPlots
AnalysisofIncompleteDataSets
8.1StatisticalAnalysiswithMissingData
8.2MissingDataintheResponse
8.2.1Least-SquaresAnalysisforCompleteData
8.2.2Least-SquaresAnalysisforFilled-upData
8.2.3AnalysisofCovariance-Bartlett'sMethod
8.3MissingValuesintheX-Matrix
8.3.1MissingVaiuesandLossinEfficiency
8.3.2StandardMethodsforIncompleteX-Matrices
8.4MaximumLikelihoodEstimatesofMissingValues
8.5WeightedMixedRegression
8.5.1MinimizmgtheMDEP
8.5.2TheTwo-StageWMRE
RobustRegression
9.1Introduction
9.2LeastAbsoluteDeviationEstimators-UnivariateCase
9.3M-Estimates:UnivariateCase.
9.4AsymptoticDistributionsofLADEstimators
9.4.1UnivariateCase
9.4.2MultivariateCase
9.5GeneralM-Estimates
9.6TestofSignificance
10ModelsforBinaryResponseVariables
10.1GeneralizedLinearModels
10.2ContingencyTables
10.2.1Introduction
10.2.2WaysofComparmgProportions
10.2.3SamplinginTwo-WayContingencyTables
10.2.4LikelihoodFunctionandMaximumLikelihoodEsti-
mates
10.2.5TestingtheGoodnessofFit
10.3GLMforBinaryResponse
10.3.1LogitModels
10.3.2LoglinearModels
10.3.3LogisticRegression
10.3.4TestingtheModel
10.3.5DistributionFunctionsasLinkFunction
10.4LogitModelsforCategoricalData
10.5GoodnessofFit-Likelihood-RatioTest
10.6LoglinearModelsforCategoricalVariables
10.6.1Two-WayContingencyTables
10.6.2Three-WayContingencyTables
10.7TheSpecialCaseofBinaryResponse
10.8CodingofCategoricalExplanatoryVariables
10.8.1DummyandEffectCoding
10.8.2CodingofResponseModels
10.8.3CodingofModelsfortheHazardRate
AMatrixAlgebra
A.lIntroduction
A.2TraceofaMatrix
A.3DeterminantofaMatrix
A.4InverseofaMatrix
A.5OrthogonalMatrices
A.6RankofaMatrix
A.7RangeandNullSpace
A.8EigenvaluesandEigenvectors
A.9DecompositionofMatrices
A.10DefiniteMatricesandQuadraticForms
A.llIdempotentMatrices
A.l2GeneralizedInverse
A.13Projectors
A.14FunctionsofNormallyDistributedVariables
A.15DifferentiationofScalarFunctionsofMatrices
A.16MiscellaneousResults,StochasticConvergence
Tables
References
Index

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