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1、AnewcharacterizationofbivariatecopulasFabrizioDuranteandPiotrJaworski∗DepartmentofKnowledge-BasedMathematicalSystemsJohannesKeplerUniversity,A-4040Linz,Austriae-mail:fabrizio.durante@jku.atInstituteofMathematics,WarsawUniversityBanacha2,02-097Warszawa,Polan
2、de-mail:jwptxa@mimuw.edu.plReceived:December15,2008;Revised:June24,2009AbstractAnewcharacterizationofbivariatecopulasisgivenbyusingthenotionofDinideriva-tives.Severalexamplesillustratetheusefulnessofthisresult.Keywords:Copula,2–increasingproperty,Dinideriva
3、tive.AMSSubjectClassification:60E05,62H20,26A24.1IntroductionA(bivariate)copulaisadistributionfunctiononI2:=[0,1]2whoseunivariatemarginsareuniformlydistributed.Specifically,C:I2→Iisacopulaifitsatisfiesthefollowingproperties:(C1)C(x,0)=C(0,x)=0foreveryx∈I,i.e.C
4、isgrounded,(C2)C(x,1)=C(1,x)=xforeveryx∈I,(C3)Cis2–increasing,thatis,foreveryx1,y1,x2,y2∈I,x1≤x2andy1≤y2,itholdsC(x1,y1)+C(x2,y2)≥C(x1,y2)+C(x2,y1).∗Correspondingauthor.Tel.4822-554-45-23,Fax.4822-554-43-001Recently,copulashavereceivedagreatpopularityduetot
5、hecelebratedSklar’sTheorem,statingthateveryjointdistributionfunctionofapairofcontinuousrandomvariablescanberepresentedbymeansofasuitablecopulaanditsunivariatemarginals.Justtohaveanideaaboutcopulatheoryand(someof)itsapplications,werefertoSchweizerandSklar(20
6、06);Joe(1997);Nelsen(2006);McNeiletal.(2005);Salvadorietal.(2007).Thegrowingimportanceofcopulasforconstructingstatisticalmodelshasoriginatedsev-eralmethodsforgeneratingnewcopulas.Thefinalgoaloftheseinvestigationsistoobtainmoreflexiblefamiliesofbivariatedistri
7、butionfunctions,havingavarietyofinterestingprop-ertiesliketaildependencies,asymmetries,widerangeofassociation.Fromanabstractviewpoint,generatingnewcopulasconsistsofdefiningafunctionC:I2→Iandprovingthatitsatisfiestherequiredproperties(C1),(C2)and(C3).Now,while
8、(C1)and(C2)areusuallytrivialtocheck,theverificationof(C3)couldbe,sometimes,atrickytaskthatmightbringus“intotheexcitinglandofcopula-exotics”(comparewithEmbrechts(2009)).Inthispaper,weshowhowthe2–increasi
