Goodness-of-Fit Tests for Symmetric Stable Distrib

arXiv:math/0602346v1[math.ST]16Feb2006Goodness-of-FitTestsforSymmetricStableDistributions–EmpiricalCharacteristicFunctionApproachMuneyaMATSUI†andAkimichiTAKEMURA‡†GraduateSchoolofEconomics,UniversityofTokyo‡GraduateSchoolofInformationScienceandTechnology,UniversityofTokyoOctober,2005AbstractWeconsidergoodness-of-fittestsofsymmetricstabledistributionsbasedonweightedintegralsofthesquareddistancebetweentheempiricalcharacteristicfunctionofthestandardizeddataandthecharacteristicfunctionofthestandardsymmetricstabledistributionwiththecharacteristicexponentαestimatedfromthedata.Wetreatαasanunknownparameter,butfortheoreticalsimplicitywealsoconsiderthecasethatαisfixed.Forestimationofparametersandthestandardizationofdataweusemaximumlikelihoodestimator(MLE)andanequivariantintegratedsquarederrorestimator(EISE)whichminimizestheweightedintegral.WederivetheasymptoticcovariancefunctionofthecharacteristicfunctionprocesswithparametersestimatedbyMLEandEISE.ForthecaseofMLE,theeigenvaluesofthecovariancefunctionarenumericallyevaluatedandasymptoticdistributionoftheteststatisticisobtainedusingcomplexintegration.Simulationstudiesshowthattheasymptoticdis-tributionoftheteststatisticsisveryaccurate.Wealsopresentaformulaoftheasymptoticcovariancefunctionofthecharacteristicfunctionprocesswithparametersestimatedbyanefficientestimatorforgeneraldistributions.1Introduction.Thefamilyofstabledistributionsisoneofthemostimportantclassesofdistributionsinprobabilitytheory.Thegeneralcentrallimittheoremassertsthatifasuitablynormalizedsumofindependentlyandidenticallydistributed(i.i.d.)randomvariableshasalimitdistribution,onlypossiblelimitsarethestabledistributions(Chapter6ofFeller(1971)).Concerningstatisticalinference,becauseoftheirattractivepropertiessuchasheavytails,manymodelsbasedonstabledistributionshavebeenconsideredinbothsocialandnaturalsciences(SamorodnitskyandTaqqu(1994),UchaikinandZolotarev(1999),RachevandMittnik(2000)).Thereforeitisimportanttoconsidergoodness-of-fittestsofstabledistributions.Howeverfewresearchesongoodness-of-fittestsofstabledistributionshavebeenconductedduetothedifficultyinexpressingtheirdensityfunctionsexplicitly.Thepurposeofthispaperistoproposegoodness-of-fittestsbasedontheempiricalcharacteristicfunction,sincethecharacteristicfunctionsofstabledistributionsareexplicitlygiven.Forpastresearchesongoodness-of-fittestsofheavy-taileddistributionsusingempiricalcharacteristicfunctionapproach,seeG¨urtlerandHenze(2000)andMatsuiandTakemura(2005).BothpaperstreatCauchy(α=1)distributionwhichisoneofthestabledistributions.Letf(x;μ,σ,α)denotethesymmetricstabledensitywiththecharacteristicfunctionΦ(t)=exp(iμt−|σt|α),1wheretheparameterspaceisΩ={−∞μ∞,σ0,0α≤2}.Hereαisthecharacteristicexponent,μisthelocationparameterandσisthescaleparameter.Forthestandardcase(μ,σ)=(0,1)wesimplywritethecharacteristicfunctionasΦ(t;α)=exp(−|t|α)andthedensityfunctionasf(x;α).Inthisparameterizationstabledistributionsformalocation-scalefamilyforeachvalueofα,i.e.,f(x;μ,σ,α)=1σf(x−μσ;α).Inordertocopewithmoregeneralsituationorfornotationalconveniencewealsowritetheparametersasθ=(θ1,θ2,θ3)=(μ,σ,α)andwritecorrespondingdensity,distributionorcharacteristicfunctionasf(x;μ,σ,α)=f(x;θ),F(x;μ,σ,α)=F(x;θ),Φ(x;μ,σ,α)=Φ(x;θ).Herewenotethatθisavector.Inthispaperweoftendifferentiatefunctionsoftheparameterθandthedataxwithrespecttox,μ,σandα.Sincewewillconsideraffineinvariant(location-scaleinvariant)tests,itisoftensufficienttoevaluatethederivativesatthestandardcase(μ,σ)=(0,1).Forexampleweusethenotationfμ(x;α)=∂∂μf(x;μ,σ,α)|(μ,σ)=(0,1)orf′(x;α)=∂∂xf(x;μ,σ,α)|(μ,σ)=(0,1).Concerningthecharacteristicfunctionwealsouse∇θΦ(t;θ)=(Φμ(t;θ),Φσ(t;θ),Φα(t;θ))where,forexample,Φμ(t;θ)=∂∂μΦ(t;μ,σ,α).Forstandardcase(μ,σ)=(0,1)wewriteΦμ(t;α)=∂∂μΦ(t;μ,σ,α)|(μ,σ)=(0,1).Givenarandomsamplex1,...,xnfromanunknowndistributionF,wewanttotestthenullhypothesisH1thatFbelongstothefamilyofstabledistributionsf(x;μ,σ,α)andthenullhypothesisH2thatFbelongstothefamilyofstabledistributionsf(x;μ,σ,α)withα=α0fixed.NotethatH1⊃H2.HereweexplainourproposedprocedurefortestingH1,becauseforH2wecansimplyreplaceˆαbyα0.Asremarkedabovestabledistributionsformalocationscalefamilyandweconsideraffineinvarianttests.Theproposedtestsarebasedonthedifferencebetweentheempiricalcharacteristicfunction(1.1)Φn(t)=Φn(t;ˆμ,ˆσ)=1nnXj=1exp(ityj),yj=xj−ˆμˆσ,ofthestandardizeddatayjandthecharacteristicfunctionwithαestimatedfromthedataΦ(t)=Φ(t;ˆα)=e−|t|ˆα.2Hereˆμ=ˆμn=ˆμn(x1,...,xn),ˆσ=ˆσn=ˆσn(x1,...,xn)andˆα=ˆαn=ˆαn(x1,...,xn)areaffineequivariantestimatorsofμ,σ,αsatisfyingˆμn(a+bx1,...,a+bxn)=a+bˆμn(x1,...,xn),ˆσn(a+bx1,...,a+bxn)=bˆσn(x1,...,xn),ˆαn(a+bx1,...,a+bxn)=ˆαn(x1,...,xn),forall−∞a∞andb0.Asequivariantestimatorsweconsidermaximumlikelihoodestimator(MLE)andanequivariantin-tegratedsquarederrorestimator(EISE)definedin(2.6)below.ThereasonforconsideringMLEisitsasymptoticefficiencyandthereasonforEISEisthatitsdefinitionissimilartoourproposedteststatistic.FollowingG¨urtlerandHenze(2000)andMatsuiandTakemura(2005)weproposethefollowingteststatistic(1.2)Dn,κ:=nZ∞−∞Φn(t)−e−|t|ˆα2w