Copulas: a Review and Recent Developments (2007)

In the bivariate case the last relation can be written asmax(u 1 + u 2 ¡ 1; 0) · C(u 1 ;u 2 ) · min(u 1 ;u 2 ):When we posses an additional information about the values of copula in the interiorof [0; 1] 2 , then the above Frechet bounds can be often narrowed, see Nelsen(1999), p. 62, Nelsen et al. (2001b) and Anjos et al. (2004).Note that only for n =2thelowerFrechet bound is a copula and the randomvariables (X; Y ) associated to min(u 1 ;u 2 )andmax(u 1 +u 2 ¡1; 0) have support in themain diagonal and the secondary diagonal of [0; 1] 2 , respectively. This means that ifF and G are the distribution functions of X and Y and P ¡ F (X) =G(Y ) ¢ =1almostsurely, the joint distribution function of (X; Y ) have associated copula min(u 1 ;u 2 )and we say that the pair (X; Y )iscomonotonic. On other hand, if P ¡ F (X) =1¡G(Y ) ¢ = 1 almost surely, the joint distribution function of (X; Y ) have associatedcopula max(u 1 + u 2 ¡ 1; 0) and the pair (X; Y )iscountermonotonic. We need thefollowing de¯nition.De¯nition (comonotonic set). The set A µ (¡1; 1) issaidtobecomonotonicif for any x = fx 1 ;::: ;x n g and y = fy 1 ;::: ;y n g in A, eitherx · y or y · x holds.The following theorem is the main result concerning comonotonicity (i.e. the bestpossible positive dependence between random vectors). Additional properties, examples,applications in Finance and Insurance can be found in Dhaene et al. (2002a,b),Theorem (characterization of comonotonicity, Dhaene et al. (2002a)). Arandom vector X = fX 1 ;::: ;X n g is comonotonic if and only if one of the followingequivalent conditions holds:(i) X has a support which is a comonotonic set;µ(ii) For all x =(x 1 ;::: ;x n ), we have H(x 1 ;::: ;x n )=min F X1 (x 1 );::: ;F Xn (x n ) ;(iii) For U » U(0; 1), we have X d = fF ¡1X 1(U);::: ;F ¡1X n(U)g;(iv) There exist a random variable Z and non-decreasing functions f i ;i=1;::: ;n;such that X = d ff 1 (Z);::: ;f n (Z)g:¥Note that in a similar way one can de¯ne a countermonotonic set andthentostate the corresponding characterization.If the true copula is assumed to belong to a parametric family fC µ ;µ 2 £g,estimates of the parameters of interests can be obtained through maximum likelihoodmethods in the context of independent and identically distributed samples. There aremainly two used methods: the fully parametric and the semiparametric one, detailed4