恰当散度非线性模型变离差的检验.doc

恰当散度非线性模型变离差的检验Vo1.27(2007)No.5数学杂志J.ofMath.(PRC)TESTINGFORVARYINGDISPERSIONINPROPERDISPERSIONNONLINEARMODELSFENGYu,LINJinguan.(1.Sc00ZofScience,NanjingUniversityofScienceandTechnology,Nanjing210094,China)(2.Dept.ofMath.,SoutheastUniversity,Nanjing210096,China)A蛞tJcfInthispaper,vgestudythetestingforvaryingdispersioninproperdispersionn0nlinearn1odels.BasedontheLRstatisticandScorestatistic,weobtainatestforheteroscedasticityinnonlinearproperdispersionmodels.Anumericalexampleisgiventoillustrateourresults.Keywords:properdispersionmodels;likelihoodratiostatistic;scorestatisticvaryingdispersion2000MRSubjectClassification:62J02Documentcode:AArticleID:02557797(2007)05'0507-061Intr0ducti0nNonconstantvarianceisacommonlyconcernedprobleminmodernregressionanalysis.Forgeneralizedlinearmodels(GLM)withexponentialfamilydistributions,sincethenorma1variancefunctionsarealwaysnonconstant(exceptinthenormalcase),itISnotnecessarytodetectnonconstantvariance.However,thevarianceproblemstillexistsforGLM,whichbecomesatestforvaryingdirpersionincludingoverdispersionandunderdispersion,1.Theproblemaboutnonconstantvariance(heteroscedasticity)goesbacktoFisher2andhasbeeninvestigatedbymanyauthorsinthepastdecade;see,r37.Inthispaper,weconsiderthetestforvaryingdispersioninproperdispersionnonlinearmodels.Alikelihoodrediostatisticandascorestatisticareobtained.Thepaperisorganzedasfollows.Section2introducestheproperdispersionnonlinearmodelsdefinedbyJorgensen(1997)8,therelevantnotationsomeinferentialresultsarealsogiven.InSection3,weobtainedtwodiagnostictestsforvaryingdispersionbasedontheLRstatisticandScorestatistic.Finally,thetwodiagnosticsmeasuresareillustrated*Receiveddate:200412-30Accepteddate:2006'0713Foundationitem:SupportedbyNSFC(10671032);SSFC(04BTJ002);thePost-DoctorialGrantinSoutheastUniversityBiography:FengYu(1964一),male,bornatNanjing,Jiangsu,Ph.D.Candidate,AssociateProfessor,majorinnonlinearregressionandstatisticsdiagnostics.E-mail:fengyunjmail,njust,edu.anwithanexampleinSection4.2ProperdispersionnonlinearmodelsSupposethattherandomvariableYl,Y2,Yareindependentandeachyihasthedensitywithform:n()÷(.),f)exp1d(.),)wherea(az)issuitableknownfunction;(.),;)isaunitdeviancedefinedonnCRistheintervalwithnopen,andsatisfyingd(y,.),)一0VyEnandVyE/a,andisdifferentiablewithrespecttouptoatleastthesecondorder.assumethat笋>0,vn.():d'一(2.1)C×n,hered(y,.),)>0Furtherwe瓦2iscalledtheunitvariancefunction.mayrepresentthemeanofthedistribution,varyrespectivelyinsubsetsCRandnR.Theparameter口一A(ACR)isreferredtoasthedispersionparameter.JorgensenE8-(1997)referredtosuchfamilyofdistributionsasaregularproperdispersionmodels(PD(,).Nowweconsiderthefollowingvaryingdispersionmodelg(/2)一f(xf;卢),yfPD(,西),i一1,z.(2.2)whereareknowmweightsandVat(Yi)=(,V().If一foralli,thenthedispersionisnominal,otherwiseitisvarying.Nowwediscussthetestfordeparturefromnominaldispersion.Itmaybereferredtoasthetestforvaryingdispersionincludingoverandunderdispersion.Forthesakeofsimplicity,weshallassumen'i=1foralli,becausen,fareknown.AnaturaltestforvaryingdispersionisHo:=,iI,z.Obviously,therearetoomanyparametersconnectedwiththemodel(2.2),sodispersionparametersareusuallymodeledbyintroducinganotherparameterytosimplifytheproblemE5-1.Fromthispointofview,weproposethefollowingvaryingdispersionmodelforproperdispersionnonlinearmodels:fYPD(,u,),=,Ig(12i)一f(zf;卢),fm(zf;y)(2.3)wherearecovariates,yisaqvectorparameterand(?;?)isaknownfunction.Inthismodel,一口一.mayberegardedasthenominaldispersionandareaddedtoreflectthevariationofdispersion.ItisassumedthatthereisauniquevalueT0ofysuchthat(;y)=1forall.Ify:T0,thenVary(y)一azV(/f)forallandthedispersionisnomina1.HencethetestforvaryingdispersionisequivalenttoatestofhypothesisHo:y=T0;Hl:yr0.(2.4)Inthispaper,weshallderiveseveralstatisticsforHobasedonthevaryingdispersionmodel(2.3).Firstweset口一(,)andthemaximumlikelihoodestimatorof口formodel(2.3)aredenotedbya=(yT,).UnderHo,themaximumlikelihoodestimatorof口isdenptedby%=(,).Further,themaximumlikelihoodestimatorof口undergivenyisdenotedby一(y)一(,(y),(y)T.3TestingforvaryingdispersionTheorem3.1Forvaryingdispersionmodels(2.3),thelikelihoodratiostatisticforNo.5Testingforvaryingdispersioninproperdispersionnonlinearmodels509H01SLR5(九)一九;(丸)一1r5一41rMs).(3.1)where5一(5(1),5(),5()一一log口(i),i=1,.1一(1,1),respectively.ProofTheloglikelihoodofmodel(2.3)canbeexpressedasz(y,一奎1n口()一1ln()一d(y;)1一一厶厶f=11一一厶厶f=1一lna(m)+lnV(yf)+Cmfd(;)5()+lnV(y)+md(,)一一1rMd+1r5+lrt(y)(3.2)whereM=diag(m1,m);d一(1,1),d(y,),5一(一lna(m),一lna(m),()一(1nV(y1),lnV(y);Then,thelikelihoodratiostaticforH0isLR一2Z(a)一l(ao)一1rMd+lrs)一lrMd+1rs).Ontheotherhand,wehave蒜=一专(1+1Ms)Sothemaximumlikelihoodestimatorsand丸respectivelymeet(1rMd).一一(1rMs).;(1rMd)口n一一(1rM)口n(3.3)Usingtheseresults,LRreducestoLR:lrs一41;)一1rs一41rMs).(3.4)NotethatifHoholds,wehavey=70andmi一1foralli,whichresultsinMI,5一5()1and;一s()1.Then(3.1)isobtainedfrom(3.4)andthetheorem3.1isproved.CorollaryFornormalandinverseGaussianmodel,(3.1)reducesto(3.5)ProofSubstituting5()一一loga()一一log4,;()一一声一into(3.1)givesLR=(一log0+1)一一log()+1)=1:log(九)+log(m),一1whichleadsto(3.5).NOWweconsiderthetestofH0usingScorestatistic.Inmodel(2.3),yistheparameterofinterestand(,)arenuisanceparameters.SowemaygetthescorestatisticforHn.Asusual,thescorestatisticforH0iSSCwhereJistheupperlertcornerof1一,l,一m,/,g一RL51OJournalofMathematicsVo1.27ofyfor口,9.Thenwehavethefollowingtheorem.Theorem3.2Forvaryingdespersionmodels(2.3),theScorecanbeexpressedasSC=丢()(+;1)P(+;1).statisticforthetestH.(3.7)Whered=(l,d)r,df(l,f).IfHoholds,themaximumlikelihoodestimatorof口aredenotedby=(,).Pistheprojeetionmatrixof一(J一11/n)and一()with=3ml/a,f一1,T/,口一1,q.ProofTogetSCfrom(3.6),weneedtofind(Ol(a)/ar)andJ(),卢).Itfollowsfromtheloglikelihoodofmodel(2.3)that一1)+(一",qOl(a)一一1(+;),().一一丢死(),Itfollowsfrom(3.3)that(1d).一一,z;()whjchresu1tsin()=一号死(J一11/n)(d+;1)Thenwehave()一一2()Nowbydirectcalculation,wecangettheFisherinformationmatrixofYfor口:(7r,),E(一):丢§丢§州1o111TMS州1o00VMO(3.8)where§一diag(;(1),;()andO=D=ao/.WhenH0holds,wehave(s)一;()J,(一J.Thenitfollowsfrom(3.8)that(jr2n)llrIl一2zS'-()()Substitutingthisresultinto(3.6)yieldsSC=軎()(+;1)()一(+;1)一去;()(+s1)P(+s1)whichresultsin(3.7).Equation(3.7)isausefulformulatodetectvaryingdispersion.Tocomputeit,oneonlyneedsthestatisticsundernullhypothesis.No.5TestingforvaryingdispersioninproperdispersionnonlinearmodelsFurthermore,thequantity(+;)p(+;)一(+s)(了,)一(+;)isjustthesumofsquaresfortheregressionof(+s)onintheconstructedmodel"d+;一岛1+"andthusitcanbeeasilyobtainedusingthestandardregressionprogram.Fromthistheoremwehaveseveralusefulresults.(1)Equation(3.7)canbeusedforcommonlyencounterednonlinearmodels.Fornormalmodel,wehaves()一一anddP=(f一),then(3.7)becomesSC一1(urP)(3.9)with一(f),=e/.ThisresultcoincideswiththeequationofCookandWeisberg(1983)E103forthetestofheteroscedasticityinnormallinearregression,asexpected.But(3.9)isalsovalidfornormalnonlinearregressioncase.ForinverseGaussiannonlinearmodelylIG(,_),wehaves()一一=,di:(一)/2and=n-(d).Ifweset=(1)一(df/a),then(3.9)stillholdswithfdefinedabove.ForgammanonlinearmodelyfGA(,ornF),wehaves()一一2log一logr(),df=2(f一f)/log(y/),andSCcanbecomputedbyformula(3.7).(2)Wemayrewrits(3.7)asastandardizedformlike(3.9).SinceE(df)=一s()一是,Var(df)=2s()andP10,ifwesetd一()withd一dfE(d)/Var专(),thenwehaveSC=(Pd)k,thatisthenonTlofthestandardizeddeviancedevaluatedatao.4AnillustrativeexampleNowweconsideranexampleastheillustrationfortheresultsofthetestforvaryingdispersion.ConsideringboththetestforheteroscedasticityinnormalnonlinearregressionandininverseGaussiannonlinearmode.WeusethemodelofMcCullaghandNelder1989E113todothestudy.ThedatarelatestoassaysonagrasshopperMelanoplussanguinipes(F).withtheinsecticidecarbofuranandthesynergistpiperonyibutoxide(PB)whichenhancesthetoxicityoftheinsecticide.Analogousto11weassumealogitlinkandnormalorinverseGassianerrorwithnonlinearpredictorgivenby一口+卢1log(x1一)+z2/(+z2)(4.1)wherez1andz2arerespectivelythedoseofinsecticideandsynergistPB.Bythealgorithm12Wei(1998),weobtaintheestimates一1,674,:2.061,西=2.896,届一1.345,and一1.708.Totestvaryingdispersion,weusemodel(4.1),thatisyfN(,mF)with(4.1),m一z,一1,15andHo:),一0.Forthistest,the1ikelihoodratiostatistic,theScorestatic.TheresultsarelistedinTable1.Table1Thetestresultsofvaryingdispersion瑶01(1)一6.635,妊.05(1)=3.841Table1givesdefiniteevidenceofheteroscedasticity.Thereforewemayapplymodel(2.3)with(4.1)512JournalofMathematicsVo1.27andIn/一tOfitthedataandexpecttOgetmorepreferableresults.ItiseasytOget7=0.5327.0.5,a一一20.566-20,and辘/iwhichvaryfromobservationtOobservation.ThevariancefunctionofmaybewrittenasVar(y')一一.,l(20_),whichisinverselyprotortionaltOAsanalternative.weusemodel(4.1)withinverseGaussianerrortOfitthedata.BoththeteststatisticsareshowninTable1.andgivedefiniteevidenceofvaryingdispersion.Somidel(2.3)withml=zmaymoreadequatelyfitthedata.Inthiscasewehave7=0.92,=0.0032andm'=xo.approximatelyequaltOthecovarlateiitself,thatiSm'i.References:1SimthP.J.,HeitjanD.F.TestingandadjustingfordeparturesfromnominaldispersioningeneralizedlinearmodelsJ.AppliedStatistics,1993,42(1):3141.2334353637383931O1112FisherR.A.ThesignificanceofdeviationsfromexpectationinaPoissonseriesJ.Biometrics,1950,6(1):17-24.SmythG.K.GeneralizedlinearmodelswithvaryingdispersionJ.J.Roy.Statist.Soc.Ser.B?1989,51(1):476O.DeanC.B.TestingforoverdispersioninPoissionandbinmialregressionmodelsJ.J.Amer.Statist.Assoc.1992,87(3):451-457.GanioA.R.,SchaffeD.w.DiagnosticsforoverdispersionJ.J.Amer.Statist.Assoc.1992,87(4):795-804.LinJ.G.,WeiB.C.TestingforheteroscedasticityinnonlinearregressionsJ.CommunicationsinStatistics,TheoryandMethods?2003,32(1):171192.LinJ.G.,WeiB.C.TestingforheteroscedasticityandcorrelationinnonlinearmidelswithcorrelatederrorsJ-.CommunicationsinStatistics,TheoryandMethods,2004,33(2):243250.Jorgensen&.ThetheoryofdispersionmodelsM.London,ChapmanandHall,1997.CoxD.R.,Hinkley.TheoreticalStatisticsM.London:ChapmanandHall,1974.CookR.D.,WeisbergS.DiagnosticsforheteroscedasticityinregressionM.Biometrika,1983,70;13O.McCullaghP.,NelderJ.A.GeneralizedLinearModelsM.London:ChapmanandHall,1989.WeiB.C.ExponentialfamilynonlinearmodelsM.SingaporetSpringer-Verlag,1998.恰当散度非线性模型变离差的检验冯予,林金官(1.南京理工大学理学院统计系,江苏南京210094)(2.东南大学数学系,江苏南京210096)摘要:本文研究了恰当散度非线性模型变离差的检验问题.基于似然比统计量和得分统计量?得到变离差的检验.并且用数值例子说明方法是有效的.关键词;恰当散度模型;似然比统计量;得分统计量;变离差MR(2000)主题分类号:62J02中图分类号:O212.1