Smoothing spline models with correlated random errors

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1、DEPARTMENTOFSTATISTICSUniversityofWisconsin1210WestDaytonSt.Madison,WI53706TECHNICALREPORTNO.966August1996SmoothingSplineModels1WithCorrelatedRandomErrorsbyYuedongWang1ResearchsponsoredinpartbyNIHGrantsR01EY09946,P60DK20572andP30HD18258.SmoothingSplineModelsWithCorrelatedRandomErrorsyYued

2、ongWangDepartmentofBiostatistics,UniversityofMichigan,AnnArbor,Michigan48109,U.S.A.August30,1996AbstractSplinesmoothingisapopularmethodofestimatingthefunctionsinanonpara-metricregressionmodel.Itsperformancegreatlydependsonthechoiceofsmoothingparameters.Manymethodsofselectingsmoothingparam

3、eterssuchasCV,GCV,UBRandGMLaredevelopedundertheassumptionofindependentobservations.Theyfailbadlywhendataarecorrelated.Inthispaper,weassumeobservationsarecorrelatedandthecorrelationmatrixdependsonaparsimonioussetofparameters.WeextendtheGML,GCVandUBRmethodstoestimatethesmoothingparametersan

4、dthecorre-lationparameterssimultaneously.Wealsoconnectasmoothingsplinemodelwiththreemixed-eectsmodels.Theseconnectionsshowthatthesmoothingsplineestimateseval-uatedatdesignpointsareBLUPestimatesandtheGMLestimatesofthesmoothingparametersandthecorrelationparametersareREMLestimates.Theseconn

5、ectionsalsosuggestawaytotasplinemodelwithcorrelatederrorsusingtheexistingSASprocedureprocmixed.Weillustrateourmethodswithapplicationstotwotimeseriesdatasetsandaspatialdataset.SimulationsareconductedtoevaluateandcomparetheperformanceoftheGML,GCV,UBRmethodsandthemethodproposedinDiggleandHu

6、tchinson(1989).TheGMLmethodisrecommendedsinceitisstableandworkswellinallsimulations.Itperformsbetterthanothermethods,especiallywhenthesamplesizeisnotlarge.Keywords:Bestlinearunbiasedprediction;Generalizedcross-validation;Gener-alizedmaximumlikelihood;Unbiasedrisk;Mixed-eectsmodel;Restric

7、tedmaximumlikelihood;Smoothingparameters;Smoothingspline;SmoothingsplineANOVA.1IntroductionInthispaperweconsiderthegeneralsmoothingsplinemodelsinWahba(1990).ForanddarbitraryindexsetT(e.g.,T=f1;


;Ng,T=[0;1]orT=E,whereEistheEuclideanyE-mail:yuedong@umich.edu.1d-spac