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1、8StructuringDependenceusingCopulaFunctions*TheLatinwordcopuladenoteslinkingorconnectingbetweenparts.Thiswordhasbeenadoptedinstatisticstodenoteaclassoffunctionsallowingtobuildcross-dependentmultivariatedistributions.Althoughthetermscorrelationanddepen-dencea
2、reoftenusedinterchangeably,theformerisaratherparticularkindofdepen-dencemeasurebetweenrandomvariables.Assuch,itsuffersfrominconveniencesduetoitslimitationincapturingotherformsofdependence.Forinstance,itisnotdifficulttofindexamplesofdependentvariablesdisplayin
3、gzerocorrelation.Theproblemofmodelingdependencestructuresisthatthisfeaturedoesnotalwaysshowoutofthejointdistributionfunctionunderconsideration.Itwouldbeofsomehelptoseparatethestatisticalpropertiesofeachvariablefromtheirdependencestructure.Copulafunctionspro
4、videuswithaviablewaytoachievethisgoal.Thischapterisorganizedasfollows.Section8.1introducesthenotionofcopulaandrelateddefinitions.Section8.2presentsanoverviewofmajorconceptsofde-pendenceandexaminestheirlinktocopulas.Sections8.3and8.4exhibitthemostimportantfam
5、iliesofcopulastogetherwiththeirproperties.Section8.5isdevotedtothestatisticalinferenceofcopulafunctions.Section8.6discussesMonteCarlosimulationtechniques.Section8.7concludeswithafewremarksandcomments.8.1CopulaFunctionsAcopulaisamathematicalfunctionCrepresen
6、tingajointdistributionFasafunc-tionofthecorrespondingmarginaldistributionsFj,j=1,...,n,i.e.,F(x1,...,xn)=C(F1(x1),...,Fn(xn)).Webeginwithadefinitionandthenpresentatheo-reticalresultstatingtheclaimedproperty.Definition(Copulafunction)Ann-copula(function)isarea
7、l-valuedfunctionCfromtheunitcube[0,1]nontotheunitinterval[0,1],suchthat:(1)Groundness:C(u)=0ifatleastonecoordinateujiszero;∗withDavideMeneguzzo.2328StructuringDependenceusingCopulaFunctions(2)Reflectiveness:C((1,...,1,uj,1,...,1))=uj;(3)N-increasingproperty:
8、Forallu1=(u1,...,u1n)andu2=(u2,...,u2n)in[0,1]n11withu1≤u2,i.e.,u1≤u2foralli,theC-volumeofthehypercubewithcornersiiu1andu2ispositive,i.e.,1···(−1)i1+i2+···+inCui1,...,uin≥0.1ni1=1,2in=1,2Itiseasyto
