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R.WERON(Wroclaw)PERFORMANCEOFTHEESTIMATORSOFSTABLELAWPARAMETERSAbstract:Inthispaper,wediscusstheissueofestimationoftheparametersofstablelaws.Wepresentanoverviewoftheknownmethodsandcomparethemonsamplesofdi erentsizesandfordi erentvaluesoftheparameters.Performancetablesareprovided.1IntroductionTheCentralLimitTheorem,whicho ersthefundamentaljusti cationforap-proximatenormality,pointstotheimportanceof {stable(sometimescalledstable)distributions:theyaretheonlylimitinglawsofnormalizedsumsofin-dependent,identicallydistributedrandomvariables.Gaussiandistributions,thebestknownmemberofthestablefamily,havelongbeenwellunderstoodandwidelyusedinallsortsofproblems.However,theydonotallowforlarge uc-tuationsandarethusinadequateformodelinghighvariability.Non-Gaussianstablemodels,ontheotherhand,donotsharesuchlimitations.Ingeneral,theupperandlowertailsoftheirdistributionsdecreaselikeapowerfunction.Inliterature,thisisoftencharacterizedasheavyorlongtails.Inthelasttwoorthreedecades,datawhichseemto tthestablemodelhavebeencollectedin eldsasdiverseaseconomics,telecommunications,hydrologyandphysics.Thispaperisdividedintosectionswhich,wehope,willguidethereaderfromtheory,throughsimulationtoestimationofparameters.Section2isanintro-ductiontothestablefamily.Weexplainthedi erencesbetweenthemostoftenused,inliterature,representationsoftheskewedstablecharacteristicfunctionandtheconfusionaroundit.Section3isaguidetosimulationof {stableran-domvariables.Theequalityinlawofaskewedstablevariableandafunctionoftwoindependentuniformandexponentialvariables(Theorem3.1)isdiscus-sed.Thesectionisclosedbyadiscussionofsomeminorerrorsinthisformulafoundindi erentpublications.Section4isconcernedwiththeissueofesti-mationoftheparametersofstablelaws.Wepresentanoverviewoftheknownmethodsandcomparethemonsamplesofdi erentsizesandfordi erentvaluesoftheparameters.BasingontheresultsofSection3weareabletocomparetheperformanceoffourmostoftenused(Fama{Roll’s,McCulloch’s,momentsandregression)estimatorsofstablelawparameters.Throughoutthepaper,wehavetriedtomakethisexcitingmaterialeasilyaccessibletoresearchersandpractitioners.Wehopethatwehaveaccomplishedthis.|||||||||||||{AMS1991subjectclassi cations.Primary60E07,62G07.Keywordsandphrases.Stabledistributions,parameterestimation.12 {StableDistributionThenotionofstableprobabilitylawwasintroducedbyL evy(1924)duringhisearlyinvestigationsofthebehaviorofsumsofindependentrandomvariables.Theambiguousnamestablehasbeenassignedtothesedistributionsbecauseasumoftwoindependentrandomvariableshavingastabledistributionwithindex isagainstablewiththesameindex .Thestabledistributioncanbemostconvenientlydescribedbyitscharacte-risticfunction(cf).Thefollowingformulaisderivedfromtheso{calledL evyre-presentationofthecfofanin nitelydivisiblelaw,giveninL evy(1934)(fordetailsseeHall(1981)).De nition2.1ArandomvariableXis {stableifandonlyifitscharac-teristicfunctionisgivenbylog (t)=8: jtj f1 i sign(t)tan 2g+i t; 6=1; jtjf1+i sign(t)2 logjtjg+i t; =1;(2.1)where 2(0;2], 2[ 1;1], 0, 2R.Since(2.1)ischaracterizedbyfourparameterswewilldenote {stabledi-stributionsbyS ( ; ; )andwriteX S ( ; ; )(2.2)toindicatethatXhasthestabledistributionwiththecharacteristicexponent(index) ,scaleparameter ,skewness andlocationparameter .When =1and =0thedistributioniscalledstandardstable.Someauthorsuseaformsimilarto(2.1),butwiththesignontheterminvo-lving reversedfor 6=1.This ispositive(negative)whenthedistributionisnegatively(positively)skewed,exceptwhen =1.Thisconfusingconventionwasusedinmanyimportantpapers,includingPress(1972a,1972b),Paulsonetal.(1975),LeitchandPaulson(1975)andKoutrouvelis(1980,1981).Thecanonicalrepresentation(2.1)hasonedisagreeablefeature.Thefunc-tions (t)arenotcontinuousfunctionsoftheparametersdeterminingthem,theyhavediscontinuitiesatallpointsoftheform =1; 6=0.However,asZolotarev(1986)remarks,setting 1=8: + tan 2; 6=1; ; =1;(2.3)yieldstheexpressionlog (t)=8: fjtj it (jtj 1 1)tan 2g+i 1t; 6=1; jtjf1+i sign(t)2 logjtjg+i 1t; =1;(2.4)2whichisafunctionjointlycontinuousin and .Thedrawbackofthisformisthat 1doesnolongerhavethenaturalinterpretationasalocationparameter.Mostauthors,therefore,usetheform(2.1)ofthecf.Anotherformofthecf,whoseusecanbejusti edbyconsiderationsofananalyticnature(seeZolotarev(1986)),isthefollowing.De nition2.2ArandomvariableXis {stablei itscharacteristicfunc-tionisgivenbylog (t)=8: 2jtj expf i 2sign(t) 2K( )g+i t; 6=1; 2jtjf 2+i 2sign(t)logjtjg+i t; =1;(2.5)whereK( )= 1+sign(1 )=8: ; 1; 2; 1:(2.6)Theparameters 2and 2arerelatedto and ,fromtherepresentation(2.1),asfollows.For 6=1, 2issuchthattan 2 K( )2!= tan 2;(2.7)andthenewscaleparameter 2= 1+ 2tan2 2 1=(2 ):(2.8)For =1, 2= and 2=2 .Theprobabilitydensityfunctions(pdf)ofstablerandomvariablesexistandarecontinuousbut,withafewexceptions,theyarenotknowninclosedform.Theexceptionsare theGaussiandistribution:S2( ;0; )=N( ;2 2), theCauchydistribution:S1( ;0; ), theL evydistributions:S1=2( ;1; ),S1=2( ; 1; ).3ComputerGenerationof {StableRandomVariablesThecomplexityoftheprob
本文标题:R. WERON (Wroclaw) PERFORMANCE OF THE ESTIMATORS O
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