Copulas: a Review and Recent Developments (2007)

(1999)) even in the cases where the associated distributions are discontinuous. Besides,the TVaR is characterized as a solution of a optimizing problem. The basiccontribution in Rockfellar and Uryasev (2000, 2002) is the following. They present apractical method of optimization to evaluate TVaR and VaR simultaneously whichis convenient in evaluation derivatives, (options, futures), market, operational creditand risk, ¯nancial risk, etc. Note that TVaR uses only the upper tail distributioninformation. Nevertheless, TVaR is not adequate to extreme events, i.e. in the caseof \low frequencies and high losses". Interesting applications of the above theorem incase of TVaR and for a class of distortion risk measures suggested by Wang (2000)and discussed by Darkiewicz et al. (2005) and Goncalves et al. (2005).3 Several new methods of copula modellingIn this section we discuss three topics related to our current research and show someof results obtained.3.1 Order statistics copulaThere are very few results in literature relating the order statistics and associatedcopulas. The random variables max(X; Y )andmin(X; Y ) are the order statistics forX and Y . Then, P (max(X; Y ) · t) =C(F (t);G(t)), e.g. Nelsen (1999), p. 25, andP (min(X; Y ) · t) =F (t)+G(t) ¡ C(F (t);G(t)):The above relations are generalized by Georges et al. (2001) as follows: Let(X 1 ;::: ;X n )beasetofcontinuousrandomvariableswithF Xi (x) =P (X i · x); i=1; 2;::: ;n: Denote by C n the associated copula and let X r:n be r-th order statistic(1 · r · n). Then its distribution function F r:n (t) =P (X r:n · t) isgivenbynXF r:n (t) = (¡1) k¡l k()l X #C n (v 1 ;::: ;v n ) ;" kXk=r l=rwhere P denotes summation over the set((v 1 ;::: ;v n ) 2 [0; 1] n j v i 2fF Xi (t); 1g;)nX± f1g (v i )=n ¡ kwith ± f1g (v i )=1ifv i =1,and0otherwise. Forr = 1 the last formula givesi=1F 1:n (t) =1¡ C n (S X1 (t);::: ;S Xn (t));where C n is the survival copula and S Xi (t) =1¡ F Xi (t); i=1;::: ;n.Wealsonotethat X n:n = max(X 1 ;::: ;X n ) and its distribution function is the diagonal section ofthe multivariate distribution F n:n (t) =C n (F X1 (t);:::;F Xn (t)).19