R. WERON (Wroclaw) PERFORMANCE OF THE ESTIMATORS O

R.WERON(Wroclaw)PERFORMANCEOFTHEESTIMATORSOFSTABLELAWPARAMETERSAbstract:Inthispaper,wediscusstheissueofestimationoftheparametersofstablelaws.Wepresentanoverviewoftheknownmethodsandcomparethemonsamplesofdierentsizesandfordierentvaluesoftheparameters.Performancetablesareprovided.1IntroductionTheCentralLimitTheorem,whichoersthefundamentaljusticationforap-proximatenormality,pointstotheimportanceof{stable(sometimescalledstable)distributions:theyaretheonlylimitinglawsofnormalizedsumsofin-dependent,identicallydistributedrandomvariables.Gaussiandistributions,thebestknownmemberofthestablefamily,havelongbeenwellunderstoodandwidelyusedinallsortsofproblems.However,theydonotallowforlargeuc-tuationsandarethusinadequateformodelinghighvariability.Non-Gaussianstablemodels,ontheotherhand,donotsharesuchlimitations.Ingeneral,theupperandlowertailsoftheirdistributionsdecreaselikeapowerfunction.Inliterature,thisisoftencharacterizedasheavyorlongtails.Inthelasttwoorthreedecades,datawhichseemtotthestablemodelhavebeencollectedineldsasdiverseaseconomics,telecommunications,hydrologyandphysics.Thispaperisdividedintosectionswhich,wehope,willguidethereaderfromtheory,throughsimulationtoestimationofparameters.Section2isanintro-ductiontothestablefamily.Weexplainthedierencesbetweenthemostoftenused,inliterature,representationsoftheskewedstablecharacteristicfunctionandtheconfusionaroundit.Section3isaguidetosimulationof{stableran-domvariables.Theequalityinlawofaskewedstablevariableandafunctionoftwoindependentuniformandexponentialvariables(Theorem3.1)isdiscus-sed.Thesectionisclosedbyadiscussionofsomeminorerrorsinthisformulafoundindierentpublications.Section4isconcernedwiththeissueofesti-mationoftheparametersofstablelaws.Wepresentanoverviewoftheknownmethodsandcomparethemonsamplesofdierentsizesandfordierentvaluesoftheparameters.BasingontheresultsofSection3weareabletocomparetheperformanceoffourmostoftenused(Fama{Roll’s,McCulloch’s,momentsandregression)estimatorsofstablelawparameters.Throughoutthepaper,wehavetriedtomakethisexcitingmaterialeasilyaccessibletoresearchersandpractitioners.Wehopethatwehaveaccomplishedthis.|||||||||||||{AMS1991subjectclassications.Primary60E07,62G07.Keywordsandphrases.Stabledistributions,parameterestimation.12{StableDistributionThenotionofstableprobabilitylawwasintroducedbyLevy(1924)duringhisearlyinvestigationsofthebehaviorofsumsofindependentrandomvariables.Theambiguousnamestablehasbeenassignedtothesedistributionsbecauseasumoftwoindependentrandomvariableshavingastabledistributionwithindexisagainstablewiththesameindex.Thestabledistributioncanbemostconvenientlydescribedbyitscharacte-risticfunction(cf).Thefollowingformulaisderivedfromtheso{calledLevyre-presentationofthecfofaninnitelydivisiblelaw,giveninLevy(1934)(fordetailsseeHall(1981)).Denition2.1ArandomvariableXis{stableifandonlyifitscharac-teristicfunctionisgivenbylog(t)=8:jtjf1isign(t)tan2g+it;6=1;jtjf1+isign(t)2logjtjg+it;=1;(2.1)where2(0;2],2[1;1],0,2R.Since(2.1)ischaracterizedbyfourparameterswewilldenote{stabledi-stributionsbyS(;;)andwriteXS(;;)(2.2)toindicatethatXhasthestabledistributionwiththecharacteristicexponent(index),scaleparameter,skewnessandlocationparameter.When=1and=0thedistributioniscalledstandardstable.Someauthorsuseaformsimilarto(2.1),butwiththesignontheterminvo-lvingreversedfor6=1.Thisispositive(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,setting1=8:+tan2;6=1;;=1;(2.3)yieldstheexpressionlog(t)=8:fjtjit(jtj11)tan2g+i1t;6=1;jtjf1+isign(t)2logjtjg+i1t;=1;(2.4)2whichisafunctionjointlycontinuousinand.Thedrawbackofthisformisthat1doesnolongerhavethenaturalinterpretationasalocationparameter.Mostauthors,therefore,usetheform(2.1)ofthecf.Anotherformofthecf,whoseusecanbejustiedbyconsiderationsofananalyticnature(seeZolotarev(1986)),isthefollowing.Denition2.2ArandomvariableXis{stableiitscharacteristicfunc-tionisgivenbylog(t)=8:2jtjexpfi2sign(t)2K()g+it;6=1;2jtjf2+i2sign(t)logjtjg+it;=1;(2.5)whereK()=1+sign(1)=8:;1;2;1:(2.6)Theparameters2and2arerelatedtoand,fromtherepresentation(2.1),asfollows.For6=1,2issuchthattan2K()2!=tan2;(2.7)andthenewscaleparameter2=1+2tan221=(2):(2.8)For=1,2=and2=2.Theprobabilitydensityfunctions(pdf)ofstablerandomvariablesexistandarecontinuousbut,withafewexceptions,theyarenotknowninclosedform.TheexceptionsaretheGaussiandistribution:S2(;0;)=N(;22),theCauchydistribution:S1(;0;),theLevydistributions:S1=2(;1;),S1=2(;1;).3ComputerGenerationof{StableRandomVariablesThecomplexityoftheprob