AN EMPIRICAL LIKELIHOOD RATIO TEST FOR NORMALITY

DepartmentofEconomicsEconometricsWorkingPaperEWP0401ISSN1485-6441ANEMPIRICALLIKELIHOODRATIOTESTFORNORMALITYLaurenBinDong&DavidE.A.GilesDepartmentofEconomics,UniversityofVictoriaVictoria,B.C.,CanadaV8W2Y2February,2004AuthorContact:LaurenDong,StatisticsCanada;e-mail:[e-mail];FAX:[phone]DavidGiles,Dept.ofEconomics,UniversityofVictoria,P.O.Box1700,STNCSC,Victoria,B.C.,CanadaV8W2Y2;e-mail:[e-mail];FAX:[phone]AbstractTheempiricallikelihoodratio(ELR)testfortheproblemoftestingfornormalityisderivedinthispaper.ThesamplingpropertiesoftheELRtestandfourothercommonlyusedtestsareprovidedandanalyzedusingtheMonteCarlosimulationtechnique.ThepowercomparisonsagainstawiderangeofalternativedistributionsshowthattheELRtestisthemostpowerfulofthesetestsincertainsituations.Keywords:Empiricallikelihood,MonteCarlosimulation,normalitytesting,size,powerJELClassifications:C12,C151INTRODUCTIONThepurposeofthispaperistodevelopanempiricallikelihoodapproachtotheproblemoftestingfornormalityinapopulation.Themaximumempiricallikelihood(EL)methodisarelativelyrecentlydevelopednonparametrictechnique(Owen,1988)forconductingestimationandhypothesistesting.Itisadistribution-freemethodthatstillincorporatesthenotionsofthelikelihoodfunctionandthelikelihoodratio.Ithasseveralmerits.First,itisabletoavoidmis-specificationproblemsthatcanbeassociatedwithparametricmethods.Second,usingtheempiricallikelihoodmethodenablesustofullyemploytheinformationavailablefromthedatainanasymptoticallyefficientway.Inthispaper,aswellasdevelopinganempiricallikelihoodratio(ELR)testfornormality,weanalyzeitssamplingpropertiesbyundertakingadetailedpowercomparisonoftheELRtestandfourothercommonlyusedtests.Itiswellknownthatanormaldistributionhasskewnesscoefficientα3=0andkurtosiscoefficientα4=3.Thesampleskewnessandkurtosisstatisticsareexcellentdescriptiveandinferentialmeasuresforevaluatingnormality.Anytestbasedonskewnessorkurtosisisusuallycalledanomnibustest.Anomnibustestissensitivetovariousformsofdeparturefromnormality.Amongthecommonlyusedtestsfornormality,theJarque-Bera(1980)test(JB),D’Agostino’s(1971)Dtest,andPearson’s(1900)χ2goodnessoffittest(χ2test)areselected.Theseareallomnibustests.Usingthemseparatelygivesustheopportunityoftestingfordeparturesfromnormalityindifferentrespects.RandomdatasetsaregeneratedusingtheGausspackage(AptechSystems,2002).Ineachreplication,thesamedatasetisusedforalloftheteststhatweconsider.Thefivetests,theELR,theJB,theDtest,theχ2,andtheχ2∗(theadjustedχ2testtobedefinedinsection3.3)areallasymptotictests.Thepropertiesofthetestsinfinitesamplesareunknown,althoughsomeofthemhavereceivedsomepreviousconsiderationintheliterature.Wesimulatetheiractualsizesandcalculatetheirsize-adjustedcriticalvalues.Theseresultsallowustoundertakeapowercomparisonofthetestsatthesameactualsignificancelevels.OneexceptionistheDtest.TheactualcriticalvaluesoftheDtestaretakenfromD’Agostino(1971and1972).ThereasonforthisisgivenSection3.2.WefindthattheELRtesthasgoodpowerpropertiesanditisinvariantwithrespecttotheformoftheinformationconstraints.Theseresultsarerobustwithrespecttovariouschangesintheparametersandtotheformofthealternativehypothesis.WerecommendtheuseoftheELRtestfornormality.1Theoutlineofthispaperisasfollows.Section2describestheapproachofusingtheempiricallikelihoodmethodandtheELRtestfortheproblemoftestingfornormality.Section3discussestheconventionalteststhatweconsider.Section4outlinestheMonteCarlosimulationexperimentandprovidestheempiricalresultsofthetests.SomeofthecomputationalissuesassociatedwiththeELRtestarediscussedinSection5,andSection6providesasummaryandourconclusions.2ELRTESTThemainfocusofthissectionistoderiveanELRtest.Considerarandomdatasetofsizen:y1,y2,...,ynwhichisi.i.d.andhasacommondistributionF0(θ)thatisunknown.θistheparametervectoroftheunderlyingdistribution.Inthecontextoftestingfornormality,theparametervectorbecomesθ=(μ,σ2)0.OurinterestistotestfornormalityH0:N(μ,σ2)usingtheinformationfromthesampleandtheempiricallikelihoodapproach.2.1ELMethodTheELmethodhasmanyfavorablefeatures.First,themethodutilizestheconceptoflikelihoodfunctions,whichisveryimportant.Thelikelihoodmethodisveryflexible.Itisabletoincorporatetheinformationfromdifferentdatasourcesandknowledgearisingfromoutsideofthesampleofdata.Theassumptionoftheunderlyingdatadistributionisimportantinconstructingaparametriclikelihoodfunction.Theusualparametriclikelihoodmethodsleadtoasymptoticallybestestimatorsandasymptoticallypowerfultestsoftheparametersifthespecificationoftheunderlyingdistributioniscorrect.Theterm“best”meansthattheestimatorhastheminimumasymptoticvariance.ThelikelihoodratiotestandtheWaldtestcanbeconstructedbasedontheestimatesanddistributionalassumptionstomakeusefulinferences.Aproblemwithparametriclikelihoodinferenceisthatwemaynotknowthecorrectdistributionalfamilytouseandthereisusuallynotsufficientinformationtoassumethatadatasetisfromaspecificparametricdistributionfamily.Mis-specificationcancauselikelihoodbasedestimatestobeinefficientandinconsistent,andinferencesbasedonthewronglyspecifiedunderlyingdistributioncan