椭圆分布族及相关类的联合混合性.pdf 18页

'˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn椭圆分布族及相关类的联合混合性尹传存,朱丹曲阜师范大学统计学院，曲阜市　273165摘要：本文进一步发展了完全混合及联合混合的理论，给出了一类具有单边无穷支撑的一元分布不是联合混合的充分条件；给出论文WangandWang(2016)中关于椭圆分布族是联合混合的新证明；进一步研究了一元及多元斜线/偏斜椭圆分布族的联合混合问题。关键词：概率论;联合混合性;多元相关性;椭圆分布;斜线/偏斜椭圆分布中图分类号：O211JointMixabilityofEllipticalDistributionsandRelatedFamiliesYinChuancun,ZhuDanSchoolofStatistics,QufuNormalUniversity,Qufu273165Abstract:Inthispaper,wefurtherdevelopthetheoryofcompletemixabilityandjointmixability,weprovidedsufficientconditionsforsomeunivariatedistributionswithone-sideunboundedsupportarenotjointmixable;WepresentanalternativeprooftoaresultofWangandWang(2016)whichrelatedtothejointmixabilityofellipticaldistributionswiththesamecharacteristicgenerator.Wealsostudythejointmixabilityofslash-ellipticaldistributionsandskew-ellipticaldistributions.Further,inthemultivariateease,thecompletemixabilityandjointmixabilityofmultivariateellipticaldistributionsandmultivariateslash-ellipticaldistributionsarealsoinvestigated.Keywords:Probabilitytheory;jointmixability;multivariatedependence;ellipticaldistributions;slash/skew-ellipticaldistributions0IntroductionCompletemixabilityandjointmixabilitydescribewhetheritispossibletogenerateran-domvariables(orvectors)fromgivendistributionswithconstantsum.TheformallydefinitionofcompletemixabilityforadistributionwasfirstintroducedinWangandWang[1]andthenextendedtoanarbitrarysetofdistributionsinWang,PengandYang[2],althoughtheconcepthasbeenusedinvariancereductionproblemsearlier(seeGaffkeandRüschendorf[3],KnottFoundations:ResearchFundfortheDoctoralProgramofHigherEducationofChina(20133705110002).AuthorIntroduction:Correspondenceauthor:YinChuancun(1963-),male,professor,majorresearchdirection:RiskManagementandActuarialScience.E-mail:ccyin@mail.qfnu.edu.cn.ZhuDan(1988-),female,Ph.DCandidates，majorresearchdirection:RiskManagementandActuarialScience.E-mail:zhudanspring@163.com.-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnandSmith[4],RüschendorfandUckelmann[5]).Thepropertiesareparticularlyofinterestinquantitativeriskmanagementandoptimizationproblemsinthetheoryofoptimalcouplings,wheredependencebetweenrisksisusuallyunknownorpartiallyunknown.Throughoutthepaper,wewriteX=dYiftherandomvariables(orvectors)XandYhavethesamedistribu-tion.ForacumulativedistributionsfunctionF,wewriteX∼FtodenoteF(x)=P(X≤x).Nextweintroducetheconceptsofcompletelymixableandjointlymixabledistributions.Byconvention,allvectorswillbewritteninboldandwillbeconsideredascolumnvectors,withthesuperscript⊤fortransposition.Definition1.1(WangandWang[1]).AunivariatedistributionfunctionFiscalledn-completelymixable(n-CM)ifthereexistnrandomvariablesX1,···,XnidenticallydistributedasFsuchthatP(X1+···+Xn=nk)=1,(0.1)forsomek∈R.AnysuchkiscalledacenterofF.IfFhasfinitefirstmomentµ,thenk=µ.Definition1.2(Wang,PengandYang[2]).TheunivariatedistributionfunctionsF1,···,Fnarejointlymixable(JM)ifthereexistnrandomvariablesX1,···,Xnwithdistributionfunc-tionsF1,···,Fn,respectively,suchthatP(X1+···+Xn=C)=1,(0.2)forsomeC∈R.Obviously,n-tupleF1,···,FnareJMdistributionswhenF1=···=Fn=F,thenFisn-CM.1PreliminaryResultsForabriefhistoryoftheconceptofthecompletemixability,werefertotherecentpapersofWang[6]andWangandWang[7].ExistingresultsoncompletemixabilityandjointmixabilityaresummarizedinWangandWang[1],Puccetti,WangandWang[8]:Thesetofalln-CMdistributionscanbecompletelydescribedonlywhenn=1orn=2.Theclassofall1-CMdistributionsconsistsofalldegeneratedistributions,whiletheclassof2-CMdistributionsconsistsofallthesymmetricdistributions,i.e.X∼Fanda−X∼Fforsomeconstanta.Someexamplesofn-CMdistributionsincludethedistributionofaconstant(forn≥1),uniformdistributions(forn≥2),normaldistributions(forn≥2),Cauchydistributions(forn≥2),binomialdistributionsB(d,p/q)withp,q∈N(ford=q);seeProposition2.3inWangandWang[1].Moreover,theyalsoprovedthatforanydistributionFwhichadmitsamonotonedensityonitsessentialsupport[a,b]withmeanµ,thenFisn-CMifandonlyifb−ab−aa+≤µ≤b−.nn-2- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnAnycontinuousdistributiononaboundedinterval(a,b)havingaconcavedensityisn-CMforanyn≥3(seePuccetti,WangandWang[8])and,anycontinuousdistributionFonaboundedinterval(a,b)admitsadensityfsatisfyingf(x)≥3on[a,b],thenFisn-CMn(b−a)(seePuccetti,WangandWang[9]).Wang,PengandYang[2]findthenecessaryandsufficientconditionsforthejointmixabilityofnormaldistributions,andrecently,WangandWang[7]obtainnecessaryandsufficientconditionsforthejointmixabilityofsomeclassesofdistribu-tions,includinguniformdistributions,distributionswithmonotonedensities,distributionswithunimodal-symmetricdensities,andellipticaldistributionswiththesamecharacteristicgenera-tor.AspointedoutinPuccettiandWang[10],asafullcharacterizationofcompletelymixabledistributionisstilloutofreach,thereareevenlessresultsconcerningsufficientconditionsforjointmixabledistributions.TheonlyavailableonesaregivenintherecentpaperofWangandWang[7].ItwasmentionedinWang,PengandYang[2]thatitseemstobeextremelydifficulttofindgeneralsufficientconditionsforJMdistributions;Wang[11]alsoclaimsthatforn=1orn=2,thesetMn(µ)ofalln-CMdistributionswithcenterµisfullycharacterized.However,forn≥3,thefullcharacterizationonMn(µ)isstillanopenquestionandhasbeenextremelychallenging.Therestofthepaperisorganizedasfollows.InSection2wediscusstheconditionsonaresultofRüschendorfandUckelmann[5]relatedtocompletemixabilityofcontinuousdistri-butionfunctionhavingasymmetricandunimodaldensity.Sections3and4arededicatedtojointmixabilityofellipticaldistributionsandslash/skew-ellipticaldistributions,respectively.Section5extendedtheresulttotheclassofmultivariateellipticaldistributionsandmultivari-ateslash-ellipticaldistributions.SomeapplicationsrelatedtoourmainresultsarepresentinSection6.Section7drawsaconclusion.2SymmetricDistributionsItwouldbeofinteresttocharacterizetheclassofcompletelymixabledistributions.Onlypartialcharacterizationsareknownintheliterature.OneniceresultforthecompletemixabilityisgivenbyRüschendorfandUckelmann[5],WangandWang[1]citethisresultinarewrittenforminthefollowingtheorem,Wang[11]providedanewproofusingdualityrepresentation.ThepropertywasalsoextendedtomultivariatedistributionsbyRüschendorfandUckelmann[5].Theorem2.1.([5])Anycontinuousdistributionfunctionhavingasymmetricandunimodaldensityisn-CMforanyn≥2.Bydefinition,themodeormodalvalueofacontinuousprobabilitydistributionisthevaluexatwhichitsprobabilitydensityfunctionhasitsmaximumvalue,sothemodeisatthepeak.Forexample,theNormaldistribution,Student-tdistribution,LogisticDistribution,-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnExponentialPowerDistributionandCauchydistributionareunimodal;Theuniformdistribu-tiononanyinterval[a,b]ismultimodal.AprobabilitydistributionfunctionFissaidtobesymmetricaroundavalueµifandonlyifF(µ−x)=1−F(µ+x−0)forallrealnumbersx,thatisX−µand−(X−µ)havethesamedistribution.IfFiscontinuous,thenconditionF(µ−x)=1−F(µ+x−0)becomesF(µ−x)=1−F(µ+x);IfFhasadensityf,thenfissymmetricaboutµifandonlyiff(µ−x)=f(µ+x)forallrealnumbersx.Notethatcompletemixabilityandjointmixabilityisaconceptofnegativedependence(cf.PuccettiandWang[12])andnotallunivariatedistributionsFaren-CM(JM).WangandWang[7]findthenecessaryandsufficientconditionsforthejointmixabilityofsomeclassesofdistribu-tions,includinguniformdistributions,distributionswithmonotonedensities,distributionswithunimodal-symmetricdensities,andellipticaldistributionswiththesamecharacteristicgener-ator.IfthesupportsofFi(i=1,2,···,n)areunboundedfromoneside,then(F1,···,Fn)isnotJMforanyn≥1;seeRemark2.2inWangandWang[7].Somecontinuousexampleswithone-sideunboundedsupportareexponentialdistribution,gammadistribution,Weibulldistribution,log-ellipticaldistribution,Paretodistribution,andsoon.Somediscreteexampleswithone-sideunboundedsupportaregeometricdistribution,Poissondistribution,logarithmicdistribution,Pascaldistribution,negativebinomialdistribution,Polya-Aepplidistribution,andsoon.ForasetofnunivariatedistributionfunctionsF1,···,Fn,asinPuccettiandWang[10],definetheminimalvarianceproblemas{()}∑n2σ(F1,···,Fn)=infVarXi:Xi∼Fi,1≤i≤n.i=1Proposition2.1inPuccettiandWang[10]tellsusσ2(F,···,F)=0holdsifandonly1nifF1,···,FnareJM.ItfollowsfromthediscussionabovethatifthesupportsofFi(i=1,2,···,n)areunboundedfromoneside,thenσ2(F,···,F)>0.Inparticular,forF=1n1···=Fn,wehave()()∑n∑VarXi=Var(X1)n+2ρij,i=11≤i−,21≤i−1.n−1Itshouldbenotedthatdeterminationofminimumandmaximumpossiblecorrelationamongtworandomvariableshashadatheoreticalandpracticalvaluesinitsownright,which-4- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnisofgreatimportanceinempiricaldataanalysis.Weemphasizethatlowerandupperboundsforthecorrelationcoefficientactuallydependonthefamilyofmarginaldistributionsinquestion,andthatthecommonlyused[−1,1]intervalcanbeinappropriateinmanyapplications.FormoredetailsinthislinewereferthereadertoDukicandMarić[13].Theorem2.2.AssumeF1,···,F2n+1are2n+1univariatedistributionfunctionswithsym-metricdensitiesonthesameinterval[−a,a](a>0),ifF(na)≤n+1(i=1,2,···,2n+1),in+12n+1then(F1,···,F2n+1)isnotJM.ProofForanyX∼F(i=1,2,···,2n+1),theconditionsF(na)≤n+1(i=iiin+12n+11,2,···,2n+1)implythat()n2nP|Xi|>a>,i=1,2,···,2n+1.n+12n+1Itfollowsthat()2∑n+1()nananaP|X1|>,···,|X2n+1|>≥P|Xi|>−2nn+1n+1n+1i=12n>(2n+1)−2n=0.2n+1Notethat{}2∑n+1{}nanaXi̸=0⊇|X1|>,···,|X2n+1|>.n+1n+1i=1Hence()2∑n+1PXi̸=0>0.i=1Thus(F1,···,F2n+1)isnotJM.Corollary2.1.(NecessaryCondition)AssumeF1,···,F2n+1are2n+1univariatedistributionfunctionswithsymmetricdensitiesonthesameinterval[−a,a](a>0),if(F1,···,F2n+1)isJM,thenthereexistssomei(1≤i≤2n+1)suchthat()n1P|Xi|≤a>.n+12n+1ThefollowingresultextendedtheconditioninTheorem2.2from[−a,a]to(−∞,∞).Theorem2.3.AssumeF1,···,F2n+1are2n+1univariatedistributionfunctionswithsym-metricdensitiesonthesameinterval(−∞,∞),ifthereexistsa>0suchthat()nnFi(a)−Fia≥,i=1,2,···,2n+1,n+12n+1then(F1,···,F2n+1)isnotJM.-5- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnProofForanyXi∼Fi(i=1,2,···,2n+1),theconditions()nnFi(a)−Fia≥,i=1,2,···,2n+1,n+12n+1implythat([])n2nP|Xi|∈a,a>,i=1,2,···,2n+1.n+12n+1Itfollowsthat(2∩n+1{[]})2∑n+1([])nanaP|Xi|∈,a≥P|Xi|∈,a−2nn+1n+1i=1i=12n>(2n+1)−2n=0.2n+1Notethat{}2∑n+12∩n+1{[na]}Xi̸=0⊇|Xi|∈,a.n+1i=1i=1Hence()2∑n+1PXi̸=0>0.i=1Thus(F1,···,F2n+1)isnotJM.ItiswellknownthatFis2-CMifandonlyifFissymmetric(seeProposition2.3inWangandWang[1]).However,forn>2,thisstatementisnottrueingeneral.Infact,thefollowingexampletellsusthesymmetryofFdoesnotimpliedFis3-CM.ItisalsoshowsthattheunimodalityassumptiononthedensityinTheorem2.1cannotremoved.Example2.1AssumeaunivariatedistributionfunctionFhasthefollowingbimodalsymmetricdensity{2r+1x2r￿,ifx∈[−a,a],2a2r+1f(x)=0,ifx∈/[−a,a],whererisapositiveinteger.Thedistributionisgivenby0,ifx<−a,F(x)=1(x2r+1+a2r+1),if−a≤x0,ψisacharacteristici1iiiigeneratorforann-ellipticaldistribution.Then(F1,···,Fn)isJMiftheinequality∑nσi≥2max{σ1,···,σn}(3.2)i=1issatisfied.ProofAssumeXfollowsfromamultivariateellipticaldistributionX∼En(,￿,ψ),where=(µ,···,µ)⊤and￿=(σ).Here1nijn×n{σ2,ifi=j,iσij=∑1(σ2−σ2),ifk̸=i̸=j.(n−1)(n−2)kl̸=klItisstraightforwardtocheckthat￿ispositivesemidefiniteundercondition(3.2)andthesummationofallentriesin￿iszero.EachmarginalXofXhasdistributionE(µ,σ2,ψ),i1ii∑ni=1,2,···,n.Thecharacteristicfunctionofi=1Xicanbeexpressedas()∑n∑φnX(t)=expitµiψ(0),t∈R.(3.3)i=1ii=1Thatis()∑n∑φnXi(t)=expitµi.i=1i=1Hence,()∑n∑nPXi=µi=1,i=1i=1andthus(F1,···,Fn)isJM.Inparticular,whenever|ψ′(0)|<∞,wegiveamoreshorterproof.Corollary3.1.SupposethatF∼E(µ,σ2,ψ),whereµ∈R,σ>0andψisacharacteristici1iiiigeneratorforann-ellipticaldistributionwith|ψ′(0)|<∞.Then(F,···,F)isJMifthe1ninequality∑nσi≥2max{σ1,···,σn}(3.4)i=1issatisfied.-8- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnProofTakingthesameXasintheproofofTheorem3.1.If|ψ′(0)|<∞,thenthecovariancematrixexistsandisequalto′Cov(X)=−2ψ(0)￿(Cambanisetal.[15]).Sothat()∑n′⊤VarXi=−2ψ(0)en￿en=0.i=1Hence(F1,···,Fn)isJM.Thisendstheproof.ThefollowingresultisadirectconsequenceofTheorem3.1.Corollary3.2.SupposethatF∼E(µ,σ2,ψ)withψ∈￿.ThenFisn-CMforanyn≥2.1∞ProofForanyn≥2,weassumethatX∼E(,￿,ψ)withψ∈￿,where=(µ,···,µ)⊤n∞and￿=σ2(1−ρ)E+σ2ρee⊤withρ=−1,Eisn×nidentitymatrix.Itsallmarginalsnnnn−1n∑nXhavedistributionF∼E(µ,σ2,ψ),thecharacteristicfunctionofXisgivenbyi1i=1i∑φnX(t)=exp(itnµ).i=1i∑nHence,P(i=1Xi=nµ)=1andthusFisn-CM.Remark3.1.Theorem2.21inFang,KotzandNg[14]showsthatψ∈￿∞ifandonlyifX∼En(,￿,ψ)isamixtureofnormaldistributions.Somesuchellipticaldistributionsarethemultivariatenormaldistribution,themultivariateT-distribution,themultivariateCauchydistributionandtheexponentialpowerdistributionEPn(,￿,β)withβ∈(0,1];seeKano[16]andGómez-Sánchez-Manzanoetal[17].SomeellipticaldistributionslikelogisticdistributionandKotztypedistributionarenotmixtureofnormaldistributions.Remark3.2.Weremarkthatamixtureofnormaldistributionsmaybeunimodal,bimodalormultimodal,andtheconditionsforbimodalitycanbefoundinRobertsonandFryer[18].TheFinCorollary3.2issymmetricbutnotnecessarilyunimodal,sothatCorollary3.2cannotreducesfromTheorem2.1.4Slash/Skew-EllipticalDistributionsInthissection,weinvestigatejointmixabilityofslash-ellipticaldistributionsandskew-ellipticaldistributions.-9- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn4.1Slash-EllipticalDistributionsAdistributioncloselyrelatedtotheellipticaldistributionistheslashdistribution.Thisdistributioncanberepresentedasthequotientbetweentwoindependentrandomvariables,anellipticaloneandthepowerofauniformdistributionU(0,1).Moreprecisely,wesaythatarandomvariableXfollowsaslashellipticaldistributionifitcanbewrittenasZX=+µ,(4.1)1UqwhereZ∼E(0,σ2,ψ)isindependentofU∼U(0,1)andq>0istheparameterrelatedto1thedistributionkurtosis.WeusethenotationX∼SE(µ,σ2,ψ;q).Thisdistributionpresents1heaviertailsthantheellipticaldistribution,andasaconsequence,itisadistributionwithgreaterkurtosis.Similarly,wesaythatarandomvectorX∈Rphasslash-ellipticalmultivariatedistributionwithvectorlocationparameter,positivesemidefinitematrixscaleparameter￿,andtailparameterq>0,ifitcanberepresentedasZX=+,(4.2)1UqwhereZ∼Ep(0,,ψ)isindependentofU∼U(0,1)andkurtosisparameterq>0.WedenotethisasX∼SEp(,￿,ψ;q).Inparticular,whenZ∼Np(,)wegetslashnormaldistribution.Whenq→∞yieldstheellipticaldistribution.MorepropertiesofthisfamilyarediscussedinGómez,QuintanaandTorres[19].IntherecentpaperBulutandArslan[20],amatrixvariateversionoftheslashdistributionisintroduced,thenewdistributionisdefinedasascalemixtureofthematrixvariatenormaldistributionandtheuniformdistribution.Usingtherepresentation(4.1),thefollowingtheoremisaconsequenceofTheorem3.1.Theorem4.1.SupposethatF∼SE(µ,σ2,ψ;q),whereµ∈R,σ>0,ψisacharacteristici1iiiigeneratorforann-variateslash-ellipticaldistribution.Then(F1,···,Fn)isJMiftheinequality∑nσi≥2max{σ1,···,σn}i=1issatisfied.Remark4.1.Weremarkthatslash-ellipticaldistributionsarestillellipticaldistributions,sothatalltheresultsinSection3areapplicabletothisclass.Wealsoremarkthattheclassofslash-ellipticaldistributionscanbefurtherextendedbyconsideringdistributionsoftheformZX=+,h(V)whereZ∼En(0,,ψ)isindependentofV,apositiverandomvariable,andhisapositiveandmonotonicmeasurablefunctionwhichmaydependonadditionalparameterq.Forexample,Reyes,GómezandBolfarine[21]consideredthecaseh(V)=1,V∼Exp(2).Vq-10- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn4.2Skew-EllipticalDistributionsArandomvariableXhasunivariateskew-normaldistributionwithasymmetryparameterλ∈Rifitsdensityfunctionisf(x)=2ϕ(x)(λx),x∈R,whereϕanddenotetheN(0,1)densityfunctionanddistributionfunction,respectively.WedenotethisbyX∼SN(λ).Theparameterλcontrolsskewness.Itisobviousthatwhenλ=0,werecovertheN(0,1)family.Theconstructionofsuchmodelsisbasedonthefollowinglemma(seeforexampleAzzalini[22]).Lemma4.1.LetYbearandomvariablewithdensityfunctionf(x)symmetricabout0,andZarandomvariablewithabsolutelycontinuousdistributionfunctionG(x)suchthatG′(x)issymmetricabout0.Theng(x)=2f(x)G(λx),−∞X),whereXisarandomvariablewhichhastheellipiticaldistributionF∼E1(0,g,π,λ),YisarandomvariablewhichhasthedistributionπandindependentofX(Fordetails,see(4)inJupp,RegoliandAzzalini[29]).NotethatZcanberewrittenasZ=X−2X1(Y>X).ItfollowsfromCorollary3.2thatthereexistnrandomvariablesX1,···,Xnidenticallydis-tributedasXsuchthatX1+···+Xn=0foralln≥2.NotethatthesupportsofthedistributionofX1(Y>X)isunboundedfromonesideandhenceitisnotn-CMforanyn≥2.ThusthedistributionofFisnotn-CMforanyn≥2.Remark4.2.ItseemswecanguessthataslongasFisasymmetricon(−∞,∞)withunboundedsupportfromtwosides,thenFisnotn-CM.Butitiswrong.Thefollowingisacounterexample.Bytheadditivity(seeProposition2.1(3)inWangandWang[1]),ifPiscontinuousdistributiononinterval(−1,1)havinganasymmetricconcavedensityandcenteredat0,QisnormalN(0,1).Thenforanyλ∈(0,1),λP+(1−λ)Qisasymmetricandisn-CMforn≥3.5ExtensionstomultivariatedistributionsInthissectionweextenttheresultsinSection3totheclassofn-variateellipticallycontoureddistributions.Wefirstintroducesomenotions.Thenotationvec(A)denotesthe-12- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnvector(a⊤,···,a⊤)⊤,whereadenotestheithcolumnofp×nmatrixA,i=1,2,···,n.we1niuseA⊗BtodenotetheKroneckerproductofthematricesAandB;Weusetr(A)todenotethetraceofthesquarematrixAandetr(A)todenoteexp(tr(A))ifAisasquarematrix.WeusethefollowingdefinitiongiveninGupta,VargaandBodnar[30].Definition5.1.LetXbearandommatrixofdimensionsp×n.Then,Xissaidtohaveamatrixvariateellipticallycontoureddistributionifitscharacteristicfunctionhastheform⊤⊤⊤E(etr(iTX))=etr(iTM)(tr(T￿T￿)).(5.1)withT:p×n,M:p×n,￿:p×p,￿:n×n,￿≥0(positivesemidefinite),￿≥0,and:[0,∞)→R.ThisdistributionwillbedenotedbyX∼Ep;n(M,￿⊗￿,).Remark5.1.Ifn=1,wesaythatXhasavectorvariateellipticaldistribution.Itisalsocalledmultivariateellipticaldistribution.InthisspecialcasethecharacteristicfunctionofXtakesontheformof(3.1)withn=p.Theimportantspecialcaseofmatrixvariateellipticallycontoureddistributionisthematrixvariatenormaldistribution(X∼Np;n(M,￿⊗￿)),itscharacteristicfunctionis()⊤⊤1⊤E(etr(iTX))=etriTM−T￿T￿.(5.2)2Thenextlemmashowsthatlinearfunctionsofarandommatrixwithmatrixvariateellipticallycontoureddistributionhaveellipticallycontoureddistributionsalso(seeTheorem2.2inGupta,VargaandBodnar[30]).Lemma5.1.LetX∼Ep;n(M,￿⊗￿,).AssumeC:q×m,A:q×p,andB:n×mareconstantmatrices.Then,⊤⊤AXB+C∼Ep;n(AMB+C,A￿A⊗B￿B,).Thenextlemmagivesthemarginaldistributionsofamatrixvariateellipticallycontoureddistribution(seeTheorem2.9inGupta,VargaandBodnar[30]).Lemma5.2.LetX∼Ep;n(M,￿⊗￿,),andpartitionX,M,and￿asX=(X1,X2),M=(M1,M2),and[]1112￿=,2122whereX1isp×m,M1isp×m,and￿11ism×m,1≤m1,n>1issaidtohavematrixvariateslashellipticaldistribution(X∼MVSEp;n(M,,,q))if1−11X=M+2ZUq2,whereZ∼Ep;n(M,Ep⊗Eq,ψ)andU>0scalarvaluedrandomvariable,independentofZ.11HereM∈Rp×nisalocationmatrixand2and2aresquarerootsofpositivedefinitescattermatrices￿and,respectively.Usingtherepresentation(4.2)andCorollary5.1wegetthecompletemixabilityofmulti-variateslash-ellipticaldistributions.Theorem5.2.SupposethatF∼SEp(,￿,ψ;q),whereψisacharacteristicgeneratorforap×nmatrixvariateellipticallycontoureddistribution.Thenforanyn≥2,thereexistnp-dimensionalrandomvectorsX1,···,XnidenticallydistributedasFsuchthatP(X1+···+Xn=n)=1.-14- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn6TwoApplicationsWangandWang[1]foundthefollowingresultinthecaseofthedensityofF1=···=Fnismonotoneandsupportedinafiniteinterval[a,b].Theorem6.1.SupposethatF∼E(µ,σ2,ψ),whereµ∈R,σ>0satisfiesi1iiii∑nσi≥2max{σ1,···,σn},(6.1)i=1andψisacharacteristicgeneratorforann-ellipticaldistribution.Thenforanyn≥2,()∑nminEf(X1+···+Xn)=fµi,X1∼F1;···;Xn∼Fni=1wherefisaconvexfunction.ProofTheresultfollowsbyusingthesameargumentasinWangandWang(2011).Definition6.1(Log-EllipticalDistribution).TherandomvectorXissaidtohavealog-ellipticaldistributionwithparametersµandσ2iflogXhasanellipticaldistributionE(µ,σ2,ψ).WeshallwriteX∼LE(µ,σ2,ψ).11ItfollowsfromValdez,Dhaene,MajandVanduffel[31]thatifthemeanofXexists,thenitisgivenby2E(X)=eψ(−σ).InspiredbyWangandWang[1]whoconsideredtheminimumofE(X1X2...Xn),Xi∼U[0,1],wegetthefollowingtheorem.Theorem6.2.SupposethatF∼LE(µ,σ2,ψ),whereµ∈R,σ>0satisfies(6.1)andψi1iiiiisacharacteristicgeneratorforann-ellipticaldistribution.Thenforanyn≥2,∑nminE(XX...X)=ei=1i.12nX1∼F1;···;Xn∼Fn∑nProofNotethatXX...X=f(logX),wheref(x)=ex.Theresultfollowsfrom12ni=1iTheorem3.1.7ConclusionsandfutureworkWepresentanewprooftoaresultduetoWangandWang[7]onJMofellipticaldistri-butionswiththesamecharacteristicgenerator.ForanyFi(i=1,2,···,n)withunboundedsupportfromoneside,itisshownthat(F1,···,Fn)isnotJMforanyn≥1.Wealsoextentsomeresultstoaclassofmultivariateslash-ellipticaldistributionsandmultivariatevariateel-lipticallycontoureddistributions.Afullcharacterizationofcompleteorjointmixabilityisstill-15- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnopen.Inparticular,findnecessaryandsufficientconditionsforcompletemixabilityorjoin-tmixabilityofboundeddistributionsoraymmetricdistributionsormultimodaldistributionsthereisstillalotofworktodo.FurtheropenquestionsinthisfieldarecollectedinWang[6].Acknowledgements.WearegratefultoRuoduWangforinsightfulcommentsandvaluablesuggestionsonthemanuscript,especiallytheprooftoTheorem2.3.TheresearchwassupportedbytheNationalNaturalScienceFoundationofChina(No.11571198)andtheResearchFundfortheDoctoralProgramofHigherEducationofChina(No.20133705110002).参考文献（References）[1]Wang,B.andWang,R.,2011.Thecompletemixabilityandconvexminimizationproblemswithmonotonemarginaldensities.J.MultivariateAnal.102(10),1344-1360.[2]Wang,R.,Peng,L.andYang,J.,2013.Boundsforthesumofdependentrisksandworstvalue-at-riskwithmonotonemarginaldensities.FinanceStoch.17(2),395-417.[3]Gaffke,N.andRüscherndorf,L.,1981.Onaclassofextremalproblemsinstatistics.MathematischeOperationsforschungundStatistik.SeriesOptimization,12(1).123-135.[4]Knott,M.andSmith,C.S.,2006.Choosingjointdistributionssothatthevarianceofthesumissmall.J.Multivar.Anal.97,1757-1765.[5]Rüschendorf,L.andUckelmann,L.,2002.Varianceminimizationandrandomvariableswithconstantsum.in:Cuadras,etal.(Eds.),DistributionswithGivenMarginals,Kluwer,2002,211-222.[6]Wang,R.,2015.Currentopenquestionsincompletemixability.ProbabilitySurveys12,13-32.[7]Wang,B.andWang,R.,2016.Jointmixability.MathematicsofOperationsResearch41(3),808-826.[8]Puccetti,G.,Wang,B.andWang,R.,2012.Advancesincompletemixability.J.Appl.Probab.49(2),430-440.[9]Puccetti,G.,Wang,B.andWang,R.,2013.Completemixabilityandasymptoticequiv-alenceofworst-possibleVaRandESestimates.Insurance:MathematicsandEconomics53(3),821-828.[10]Puccetti,G.andWang,R.,2015b.Detectingcompleteandjointmixability.JournalofComputationalandAppliedMathmatics280,174-187.-16- 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