Copulas: a Review and Recent Developments (2007)

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such thatC(u; v) =© ru;v¡© ¡1 (u); © ¡1 (v) ¢ : (8)Note that the last theorem does not states that every copula C can be representedby a Gaussian copula, but that for each pair (u; v) 2 [0; 1] 2 always exist a memberof the family of the Gaussian copulas for which (8) holds whenever (7) is true. Infact, r u;v de¯ned by (7) is another local dependence measure related to the distancebetween the copula C and the Gaussian copula in [0; 1] 2 .4 Extreme value copulas and applicationsExtreme value multivariate modelling presents special features which rule out membersof the most well known general family of distributions, the elliptical family. Thevariables involved are usually non-exchangeable, positively associated (if not independent),with non-linear forms of dependence. In applications, additional complicationsarise from the usual scarcity of data.The asymptotic theory for extremes is well developed and univariate modelling basicallyfalls in one of the two categories: modelling standardized maxima through thegeneralized extreme value distribution (GEV), or modeling excesses over high thresholdsvia the generalized Pareto distribution (GPD). In the multivariate case thereare analogs of the two approaches, which follow from extensions of the Fisher andTippett theorem, e.g. Fisher and Tippett (1928). However, the limit family has no¯nite parameterization, see Galambos (1978), Joe (1997), Reiss and Thomas (1997)and Coles (2001). It is possible to characterize sub-families of multivariate extremevalue distributions by assuming extreme value models for the margins and specifyingparametric models for the dependence function, see Pickands (1981), Deheuvels(1978), de Haan (1985) and Tawn (1988), among others.The classical results in extreme value theory may be rewritten in terms of copulas.Let (X i;1 ;X i;2 ), i =1; 2;:::, be a sequence of random variables in (¡1; 1) 2 withjoint distribution H and marginals F Xi , i =1; 2. Let (M 1 ;M 2 )bethecomponentwisemaxima, i.e., M i = maxfX 1;i ; :::; X N;i g, i =1; 2. The bivariate version of the Fisher-Tippett theorem states that, if there exist sequences of normalizing constants a iN > 0,b iN 2 (¡1; 1), i =1; 2, such that the joint distributionµM1 ¡ b 1NP· x 1 ; M 2 ¡ b 2N· x 2 = H N (a 1N x 1 + b 1N ;a 2N x 2 + b 2N )a 2Na 1Nconverges in distribution as N !1to a proper distribution W (x 1 ;x 2 )withnondegeneratemargins, then W (x 1 ;x 2 ) is a bivariate extreme value distribution. Thismeans that H belongs to the domain of attraction of W ,denotedbyH 2 MDA(W ).28¥
Thus F Xi 2 MDA(W i ), where W i are extreme value distributions (GEV), i =1; 2.It may be shown, e.g. Resnick (1987), that W (x 1 ;x 2 ) must satisfy the max-stabilityrelation: for all N ¸ 1, there exists a iN > 0, b iN 2 (¡1; 1), i =1; 2, such thatW N (x 1 ;x 2 )=W (a 1N x 1 + b 1N ;a 2N x 2 + b 2N ):It is interesting to note that the normalizing sequences do not a®ect the marginallimiting distributions, which are unique up to a±ne transformations. Thus, in thebivariate case, normalizing sequences fa iN ;b iN g; i =1; 2,maybechosensuchthatthe marginal limits are of the sametype(eitherGumbel,Frechet, or Weibull). Thefollowing result shows that these normalizing sequences have in°uence on the a±netransformation of the marginal distributions, but do not a®ect the copula.Theorem (a±ne transformation, Charpentier (2004)). Consider (X 11 ;X 12 );:::;(X N1 ;X N2 ); ::: sequence of independent and identically distributed versions of (X 1 ;X 2 ),with joint distribution H. Assume that there are normalizing sequences a iN > 0;a 0 iN >0, b iN ;b 0 iN 2 (¡1; 1), i =1; 2, such that½ H N (a 1N x 1 + b 1N ;a 2N x 2 + b 2N ) ! W (x 1 ;x 2 );H N (a 0 1N x 1 + b 0 1N ;a0 2N x 2 + b 0 2N ) ! W 0 (x 1 ;x 2 )as N !1, for two nondegenerate distributions W (x 1 ;x 2 ) and W 0 (x 1 ;x 2 ). Thenthe marginal distributions of W (x 1 ;x 2 ) and W 0 (x 1 ;x 2 ) areuniqueuptoana±netransformation, i.e., there are ® X1 , ® X2 , ¯X1 , ¯X2 such thatW 1 (x 1 )=W 0 1 (® X 1x 1 + ¯X1 ) and W 2 (x 2 )=W 0 2 (® X 2x 2 + ¯X2 ):Further, the dependence structures of W (x 1 ;x 2 ) and W 0 (x 1 ;x 2 ) are equal, i.e. thecopula are equal, C W = C W0 . ¥Let C be the copula associated to H. From the continuity of the extreme valuedistributions it follows that there exists a unique copula C ¤ such thatDeheuvels (1978) shows thatThen C ¤ satis¯esW (x 1 ;x 2 )=C ¤ (W 1 (x 1 );W 2 (x 2 )) :C ¤ (u 1 ;u 2 )=lim N!1 C N (u 1 N1 ;u 1 N2 ) :C ¤ (u t 1;u t 2)=C t ¤(u 1 ;u 2 ) (9)for all t>0. Expression (9) de¯nes an extreme value copula.We now write the bivariate copula based version of the Fisher-Tippett theorem,see Demarta (2001).29
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