Copulas: a Review and Recent Developments (2007)

can be employed in probability theory to characterize dependence concepts and togenerate counter-examples as demonstrated by Nelsen (1995).Given a joint distribution function H with continuous marginals F X1 ;::: ;F Xn ,asin Sklar's Theorem, it is easy to construct the corresponding copula:C(u 1 ;::: ;u n )=H(F ¡1X 1(u 1 );::: ;F ¡1X n(u n )); (3)where F ¡1X iis the cadlag inverse of F Xi , i.e., F ¡1X i(u i )=supfx i jF Xi (x i ) · u i g,fori =1;::: ;n.Note as well that if X 1 ;::: ;X n are all continuous random variables with distributionfunctions as above, then C is the joint distribution function for the randomvariables U i = F Xi (X i ), i =1;::: ;n, (i.e. obtained by the probability integral transform)which are uniformly distributed on [0; 1], to be denoted further by U(0; 1).As multivariate distributions with uniform one-dimensional margins, copulas providevery convenient models for studying dependence structure with tools that arescale-free. Alternatively, one could transform the distribution of X i to any otherdistribution, but U(0; 1) is particularly easy, being parameter free. Really, if thecontinuous random variables X and Y have distribution functions F and G with correspondingdensities f X and g Y ,thenY = G ¡1 (F (X)) transforms the density f X intog Y , e.g. Rohatgi (1984), p. 460.Each copula (n-copula) represents the whole class of continuous bivariate (multivariate)distributions from which it has been obtained when one-dimensional marginalswere transformed by their distribution functions. The similar property, however, doesnot hold when the original distributions are discrete, or mixed discrete-continuous. Afterthe transformation by marginal distribution functions, the copula is not uniquelyde¯ned and consequently cannot be used for dependence studies analogously as in thecontinuous case. In Section 2.1.2 we discuss thiscopulapitfallandshowasolution.Strictly increasing transformations of the underlying random variables result inthe transformed variables having the same copula. From expression (3), we may observethat the dependence structure embodied by the copula can be recovered fromthe knowledge of the joint distribution H and its margins. Therefore, the copulaof multivariate distribution can be considered as the part describing the dependencestructure as a complement to the conduct of each of its marginals. The use of copulasis a way to solve the di±cult problem, namely ¯nding the whole multivariate distribution,by performing two simpler steps. The ¯rst one starts by modelling each marginaldistribution, and the second one consists of estimatingacopulawhichsummarizesallthe dependence structure. A candidate can be selected from some parametric copulafamily, see Joe (1997), Section 5. But, it appears another real problem - how tochoose the right copula?For every n-copula C the usual Frechet bounds are given byà nX!max u i ¡ n +1; 0 · C(u 1 ;::: ;u n ) · min(u 1 ;::: ;u n ):i=13