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1、NonparametricRegressioninRAnAppendixtoAnRCompaniontoAppliedRegression,SecondEditionJohnFox&SanfordWeisberglastrevision:13December2010AbstractIntraditionalparametricregressionmodels,thefunctionalformofthemodelisspeci edbeforethemodelis ttodata,andtheobjectistoestimatetheparam
2、etersofthemodel.Innonparametricregression,incontrast,theobjectistoestimatetheregressionfunctiondirectlywithoutspecifyingitsformexplicitly.InthisappendixtoFoxandWeisberg(2011),wedescribehowto tseveralkindsofnonparametric-regressionmodelsinR,includingscatterplotsmoothers,where
3、thereisasinglepredictor;modelsformultipleregression;additiveregressionmodels;andgeneralizednonparametric-regressionmodelsthatareanalogstogeneralizedlinearmodels.1NonparametricRegressionModelsThetraditionalnonlinearregressionmodel(describedintheAppendixonnonlinearregression)
4、tsthemodely=m(x;)+"whereisavectorofparameterstobeestimated,andxisavectorofpredictors;theerrors"areassumedtobenormallyandindependentlydistributedwithmean0andconstantvariance2.Thefunctionm(x;),relatingtheaveragevalueoftheresponseytothepredictors,isspeci edinadvance,asitisi
5、nalinearregressionmodel.Thegeneralnonparametricregressionmodeliswritteninasimilarmanner,butthefunctionmisleftunspeci ed:y=m(x)+"=m(x1;x2;:::;xp)+"fortheppredictorsx=(x1;x2;:::;xp)0.Moreover,theobjectofnonparametricregressionistoestimatetheregressionfunctionm(x)directly,rathe
6、rthantoestimateparameters.Mostmethodsofnonparametricregressionimplicitlyassumethatmisasmooth,continuousfunction.1Asinnonlinearregression,itisstandardtoassumethat"iNID(0;2).Animportantspecialcaseofthegeneralmodelisnonparametricsimpleregression,wherethereisonlyonepredictor:y
7、=m(x)+"Nonparametricsimpleregressionisoftencalledscatterplotsmoothing"becauseanimportantap-plicationistotracingasmoothcurvethroughascatterplotofyagainstx.Wefrequentlyusenonparametricregressioninthismannerinthebodyofthetext.1Anexceptiontotheimplicitassumptionofsmoothnessiswa
8、veletregression,notdiscussedinthisappendix,whichisimplementedinR,e.g.,inthe
