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