线性模型
作者：C.R.Rao/等 整理日期：2023-04-25 08:53:51
片断：
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,
arethosecharacteristicswhoseva luesintheobservationperiodcanvary.
Thosecharacteristicsthatdonotvarycanberegardedasthestructureof
therelation.Thestructureconsistsofthefunctionalformoftherelation,
includingtherelationbetweenthemainvariables,thetypeofprobabil-
itydistributionoftherandomvariables,andtheparametersofthemodei
equations.
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