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

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

y Genest et al. (1993) and Shi and Louis (1995), see also Chebrian et al. (2002). Theparametric methods relies on the assumption of parametric marginal distributions.Each parametric margin is then plugged in the full likelihood which is maximized withrespect to µ. The ¯nal results depend on the right speci¯cation of all marginals. Inthe semiparametric method, the marginal empirical cumulative distribution functionscan be incorporated in the likelihood. In this case the estimation procedure su®ersfrom the loss of e±ciency, see Genest and Rivest (1993).Besides these two methods, it is also possible to estimate a copula by some nonparametricmethods using empirical distributions, see Deheuvels (1979, 1981a,b) andFermanian et al. (2004). Non-parametric estimation of copulas in context of timedependence based on kernel approach have been studied by Fermanian and Scaillet(2003). The goodness of ¯t tests for copulas are discussed by Fermanian (2005).The copula theory has an incredible evolution during the last decade, motivatedby its application in Probability theory, Biostatistics, Finance, Insurance, Economics,see Embrechts et al. (2002), the recent monographs Cherubini et al. (2004), Nelsen(2006) and references therein. The interested reader can ¯nd many research andapplied papers visiting, say, the home pages of the following institutions and theirrelated links:² Financial and Actuarial Research Group at the ETH, Zurich, Switzerland;² Financial Econometric Research Center, UK;² Actuarial Science, Katholieke Universiteit Leuven, Belgium;² Center for Research in Economics and Statistics, France;² International Center for Financial Asset Management and Eng., Switzerland;² Research Group at Credit Lyonnais, France, etc.Therefore, there is no need to demonstrate the importance of copula theory fromtheoretical and practical point of view. In spite of this, the research on relevantspeci¯cations for copulas and on their time (dynamic) dependence is still in its infancy.In Section 2 we present brie°y two extensions of the probability integral transform,Bernstein and Bertino's family of copulas, notion about quasi-copulas, several copulalikeversions related to conditional and pseudo-conditional copulas usedintimeseriescontext. We consider as well copula applications for evaluating quantile risk measures.In Section 3 we discuss three topics: order statistics copula, copulas with multivariatemarginals and a representation of copula via a local dependence measure. In Section4 is given a review of some known facts and recent author's results treating theapplications of extreme value copulas. Several conclusions are given at the end.5

2 Extended copula conceptsIn this section we outline several important new advanced concepts which are importantcontribution into the modern copula theory and its application.2.1 Extensions of the probability integral transformHere we will present two recent generalizations of the classical probability integraltransform, i.e. F (X) » U(0; 1), where X is a continuous random variable withdistribution function F , see Feller (1968).2.1.1 Bivariate probability integral transform andrelated Kendall distribution functionsAn extension of probability integral transform is given by Nelsen et al. (2001a). Topresent it we need the following de¯nition.De¯nition (distribution function of H 1 given H 2 , Nelsen et al. (2001a)).Let H 1 and H 2 be bivariate distribution functions with common continuous marginaldistribution functions F and G. LetX and Y be random variables whose joint distributionfunction is H 2 ,andlet − ®H 1 jH 2 (X; Y ) denote the random variable H1 (X; Y ).The H 2 -distribution function of H 1 , which we denote (H 1 jH 2 ), is given byµ µ −H1 ® ©(x;(H 1 jH 2 )(t) =P jH 2 (X; Y ) · t = ¹ H2 y) 2 (¡1; 1)2¯¯H 1 (x; y) · t ªfor t 2 [0; 1], where ¹ H2 (:) denotes the measure on (¡1; 1) 2 induced by H 2 .Since copulas are bivariate distribution functions with uniform margins on [0; 1],we have an analogous de¯nition for copulas: If C 1 and C 2 are any two ® copulas, andif U; V » U(0; 1) whose join distribution function is C 2 ,thenhC 1 jC 2 (U; V ) denotesthe random variable C 1 (U; V )andtheC 2 -distribution function of C 1 is given byµ µ −C1 ® ©(u;(C 1 jC 2 )(t) =P jC 2 (U; V ) · t = ¹ C2 v) 2 [0; 1]2¯¯C1 (u; v) · t ªfor t 2 [0; 1], where similarly ¹ C2 (:) denotes the measure on [0; 1] 2 induced by C 2 .The following theorem is a bivariate analog of the probability integral transform.Theorem (bivariate probability integral transform, Nelsen et al. (2001a)).Under the above notations, let C 1 and C 2 be the copulas associated to H 1 and H 2 .Then (H 1 jH 2 )=(C 1 jC 2 ). ¥In the particular case when C 1 and C 2 are represented by min(u; v), independencecopula ¦(u; v) =uv and max(u + v ¡ 1; 0), one obtains the distribution of (C 1 jC 2 )(t)listenintheTable1.6

Table 1: Distribution of (C 1 jC 2 )(t), Nelsen et al. (2001a).C 1 =C 2 min(u; v) ¦=uv max(u + v ¡ 1; 0)min(u; v) t 2t ¡ t 2 min(2t; 1)p p¦=uvt t(1 ¡ lnt) 1 ¡ max(0; 1 ¡ 4t)max(u + v ¡ 1; 0) 0:5(t +1) 1 ¡ 0:5(1 ¡ t) 2 1Thus, for example, (min(u; v)jmin(u; v)) » U(0; 1); (min(u; v)juv) » Beta(1; 2);(min(u; v)jmax(u + v ¡ 1; 0)) » U(0; 0:5); the distribution of (uvjmin(u; v)) »Beta(0:5; 1); (max(u + v ¡ 1; 0)jmax(u + v ¡ 1; 0)) is a degenerated one, etc.If we assume additionally that the copulas C 1 and C 2 coincide, i.e. C 1 = C 2 = C,we obtain the Kendall distribution function, see Nelsen et al. (2003). If X and Y havejoin distribution function H, the Kendall distribution function of (X; Y ), to denoteit by K C , depends only on the copula C associated to H, i.e.for any t 2 [0; 1]. Formally, we haveK C (t) =P (H(X; Y ) · t) =P (C(U; V ) · t)K C (t) =¹ Hµ ©(x;y) 2 (¡1; 1)2¯¯H(x; y) · tª = ¹ Cµ ©(u;v) 2 [0; 1]2¯¯C(u; v) · tª :From Table 1, for example, one can see that only in the comonotonic case wehave K min(u;v) (t) =t, i.e. K min(u;v) » U(0; 1); K uv (t) =t(1 ¡ lnt), (which is thedistribution of the product of two U(0; 1) variates), when C is presented by theindependence copula and K max(u+v¡1;0) (t) = 1 in the countermonotonic case.In fact, Kendall distribution function is an one-dimensional summary of underlyingdependence structure of (X; Y ). We will use it again in Section 3.2.2.1.2 Grade transformation version of Sklar's theoremHere we discuss the grade transformation which, being an extension of probabilityintegral transform, enables unique copula representation of a multivariate distributionof any type for which the marginal distributions can be de¯ned.As we discussed already in Introduction, one-dimensional margins of a continuousmultivariate distribution are transformed by their distribution functions then oneobtains a copula (n-copula) determined by a joint distribution in a unit square [0; 1] 2(or n-cube [0; 1] n ) with margins having U(0; 1) distribution. It is also well known thatsuch property does not hold in a discrete case. Indeed, if the discrete random variableX takes values x 1

Therefore various attempts of \continuisation" of random variables were madeover the years, with grade transformation being one of them. This approach is verygeneral, applicable to all random variables for which their distribution functions canbe de¯ned. The so-called grade transformation, proposed by Szczesny (1991), resultsin a continuous, uniformly distributed variables and therefore can be considered anextension of probability integral transform.De¯nition (grade transformation, Szczesny (1991)). Let X be a random variablewith distribution function F (x). Then the cumulative distribution of a variableX transformed by its distribution function can be expressed asZP (F (X) · u) = I F (F (x);u)dF (x);whereI F (F (x);u)=½ 1; if F (x) · u;0; if F (x) >u:Let us now substitute I F by the randomizing transformation I ¤ Fde¯ned asI ¤ F (F (x);u)= 8

It should be mentioned here that grade transformation can be seen as analogousto a simple interpolation used by Schweitzer and Sklar (1974), and later by Nelsen(1999) as an example of a possible extension of any subcopula to a copula, whenmarginal distributions are not continuous. Due to its interesting properties, the gradetransformation is playing a special role in statistics enabling uni¯ed approach toanalysis of continuous, discrete or categorical data.2.2 Bernstein and Bertino's family of copulas2.2.1 Bernstein copulaSancetta and Satchell (2004) introduced a family of copulas de¯ned as Bernsteinpolynomials. The choice is motivated by the fact that Bernstein polynomials areclosed under di®erentiation and most of polynomial representations, e.g. Hermitepolynomials, Pad approximations, say, do not share the same properties in the contextof copula function.De¯nition (Bernstein copula, Sancetta and Satchell (2004)). Let ®( k 1m 1; ¢¢¢ ; k nbe real valued constant, k i 2f1; 2;:::g such that 1 · k i · m i ;i=1;::: ;n and substituteP ki ;m i(u i )=m i ()k i u k ii (1 ¡ u i) m i¡k i: If C Be :[0; 1] n ! [0; 1], whereC Be (u 1 ;::: ;u n )= X ¢¢¢ X µk1® ; ¢¢¢ ; k nP k1 ;mm 1 m1(u 1 ) :::P kn ;m n(u n )nk 1 k nsatis¯es the properties of the copula function from the Formal copula de¯nition, thenC Be is a Bernstein copula for any k i ¸ 1:The Bernstein copula generalizes the family of polynomial copula, which is aspecial case of copula with polynomial sections. An earlier reference for the Bernsteinpolinomial copula construction is Li et al. (1997). For more details on copulas withpolynomial sections see Nelsen (1999), Section 3.In Sancetta and Satchell (2004) are studied statistical properties in terms of bothdistributions and densities. In particular, it is shown that the coe±cients of theBernstein copula C Be have a direct interpretation as the points of some arbitraryapproximated copula C, i.e.C( k 1m 1; ¢¢¢ ; k nm n)=®( k 1m 1; ¢¢¢ ; k nm n).Sancetta and Satchell (2004) also develop a theory of approximation of multivariatedistributions in terms of Bernstein copula. They give the rate of consistency whenthe Bernstein copula density is estimated non-parametrically.The following important decomposition is achievednYC(u 1 ;::: ;u n )= u i + D(u 1 ;::: ;u n ); (u 1 ;::: ;u n ) 2 [0; 1] n ;i=1where D(u 1 ;::: ;u n ) is a perturbation term containing all information about the dependenceof (U 1 ;::: ;U n ), where U i » U(0; 1); i=1;::: ;n: This new representation9m n)

leads to a general approach in estimation as well as simpli¯cations of many operationswhenever a parametric copula is given.2.2.2 Bertino's family of copulasIn this section we present some results related to Bertino's family of copulas. In orderto do this, we need the notion of diagonal: a diagonal is a function ± :[0; 1] ! [0; 1]such that ±(0) = 0 and ±(1) = 1; ±(t) · t for all t 2 [0; 1] and for every t 1 max(u; v)] = P [min(U; V ) · t

Nelsen et al. (2003) have used Bertino family of copulas to show the followingresults relating Bertino copula and associated Kendall distribution (i.e. K C (t) =P (H(X; Y ) · t), see also Section 2.1.1) of some pair of random variables (X; Y ).)Theorem (Kendall distribution associated to Bertino copula, Nelsen et al.(2003)). Let B ± be the Bertino copula. Then for t 2 [0; 1] we have K B± =2± ¡1 (t)¡t,where ± ¡1 denotes the cadlag inverse of ±. ¥The next theorem can serve as a base for inducing the equivalence relation ´K onthe set of copulas.Theorem (existence, Nelsen et al. (2003)). Let F be a right-continuous distributionfunction such that F (0 ¡ )=0and F (t) ¸ t for all t 2 [0; 1]. Then there existacopulaC such that K C (t) =F (t) for all t 2 [0; 1]. ¥Therefore, the properties of Bertino copula can be useful in new developments incopula theory. For instance, in the above theorem the Bertino family of copulas isused to show that every distribution function satisfying the properties of the Kendalldistribution function is the Kendall distribution function of some pair of randomvariables (X,Y).2.3 Quasi-copulasThe notion of quasi-copula was introduced by Alsina et al. (1993) in order to showthat a certain class of operations on univariate distribution functions is not derivablefrom corresponding operations on random variables de¯ned on the same probabilityspace. The same concept was also used by Nelsen et al. (1996) to characterize,in a given class of operations on distribution functions, those that do derive fromcorresponding operations on random variables.Following Alsina et al. (1993), who investigated the concept in the bivariate caseonly, let a track refer to any subset B of the unit square that can be written in theform B =[(F (t);G(t)) : 0 · t · 1] for some continuous distribution functions F andG such that F (0) = G(0) = 0 and F (1) = G(1) = 1. Genest et al. (1999) characterizethe concept of quasi-copulas as follows.Theorem (characterization of quasi-copula, Genest et al. (1999)). A quasicopulais any function Q :[0; 1] 2 ! [0; 1] such that for every track B, thereexistsacopula C B that coincides with Q on B, namely Q(u; v) =C B (u; v); (u; v) 2 B, thatmeet the three following requirements:(i) Q(0;t)=Q(t; 0) = 0 and Q(t; 1) = Q(1;t)=t for all 0 · t · 1;(ii) Q is non-decreasing in each of its arguments;¥11

(iii) Q satis¯es Lipschitz's condition, i.e. for all u 1 ;u 2 ;v 1 ;v 2 2 [0; 1]jQ(u 1 ;v 1 ) ¡ Q(u 2 ;v 2 )j·ju 1 ¡ u 2 j + jv 1 ¡ v 2 j: ¥Each copula is a quasi-copula, but there exist quasi-copulas which are not copulas.Thereasonisthatcondition(iii)inlasttheorem is less restrictive than the 2-increasingcopula property (see the Formal copula de¯nition).Example (quasi-copula which is not copula). Such is the function8< uv; if 0 · v · 0:25;Q 2 (u; v) = uv + 1 (4v ¡ 1)sin(2¼u); if 0:25 · v · 0:5;:24uv + 1 (1 ¡ v)sin(2¼u); if 0:5 · v · 1;24considered by Genest et al. (1999).LetusnotethatNelsenetal. (2002b)provide a new simple characterization ofquasi-copulas in terms of absolute continuity of their vertical and horizontal sections.We know that for every copula C and any rectangle R =[u 1 ;u 2 ]£[v 1 ;v 2 ] 2 [0; 1] 2 ,the C-volume of R denoted by V C (R) is a value between [0; 1]. The next theorempresents the corresponding result for quasi-copulas.Theorem (bounds for quasi-copulas, Nelsen et al. (2004)). Let Q be a quasicopula,and R =[u 1 ;u 2 ] £ [v 1 ;v 2 ] 2 [0; 1] 2 any rectangle. Then ¡ 1 3 · V Q(R) · 1.Furthermore, V Q (R) =1if and only if R =[0; 1] 2 ,andV Q (R) =¡ 1 3 implies R =[ 1; 2 3 3 ]2 . ¥In Nelsen et al. (2004) is developed a method based on quasi-copulas in order to¯nd the best-possible bounds (improving thus the usual Frechet bounds) on bivariatedistributions functions with ¯xed marginals, when additional informations of adistribution-free nature is known.Let C be a non-empty set of copulas with a common domain D. Let C inf andC sup denote, respectively, the point-wise in¯mum and supremum of C, i.e. foreach(u; v) 2 D we de¯neC inf (u; v) =inffC(u; v) :C 2 Cg and C sup (u; v) =supfC(u; v) :C 2 Cg:The functions C inf and C sup are bounds for C since for each C 2 C we haveC inf (u; v) · C(u; v) · C sup (u; v) forall(u; v) 2 D, and are clearly point-wise bestpossible. The following theorem states that the above bounds for a non-empty set ofcopulas are also quasi-copulas.Theorem (best possible bounds, Nelsen et al. (2004)). Let C be a non-emptyset of copulas. Then C inf and C sup are quasi-copulas. ¥In a recent study, Nelsen et al. (2002a) de¯ne and show basic properties of multivariateArchimedian quasi-copulas. In particular, properties concerning generators,diagonal sections, permutation symmetry, level sets and order are examined.12

2.4 Time dependent copulasIn practice, a powerful tool to specify the underlying model are the conditional distributionswith respect to the past observations. They are more useful than the jointor marginal unconditional distributions and have become a standard tool in ¯nanceand insurance, see Embrechts et al. (2002), Cherubini et al. (2004), Patton (2006),Fermanian and Scaillet (2005), Fermanian and Wegkamp (2004). The reason is thepresence of temporal dependencies in returns of stock indices, credit spreads or interestrates of various maturities, etc.In Finance and Economics, it is often necessary to simulate the future values ofsome market factor, say S, atvariousdatest>0. Knowing the spot value S 0 today,future realizations of S t given S 0 have to be simulated. Therefore, the dependencefunction C 0;t (:jS 0 ) needs to be speci¯ed. A ¯rst attempt to formalize the functionC 0;t (:jS 0 ) for a ¯xed horizon t is due to Patton (2006) who introduced the conditionalcopula in bivariate case as follows.De¯nition (conditional bivariate copula, Patton (2006)). Let X; Y and W becontinuous random variables. The conditional copula of (X; Y )jW ,whereXjW » F Wand Y jW » G W , is the conditional joint distribution function of U = F (XjW ) »U(0; 1) and V = G(V jW ) » U(0; 1) given W .Therefore, the conditional bivariate copula is the joint distribution of two conditionallyU(0; 1) random variables. In general, the n-dimensional conditional copulais derived from any distribution function such that the conditional joint distributionof the ¯rst n variables is a copula for all values of the conditioning variables. It issimple to show that conditional copula satis¯es the properties similar to the usualcopula. The following version of Sklar's theorem holds.Sklar's Theorem (for conditional bivariate copula, Patton (2006)). Let H Wbe the joint conditional distribution function with marginal distribution of (X; Y )jW .Let F W be the conditional distribution of XjW and G W be the conditional distributionof Y jW . Assume X and Y are continuous in x and y. Then there exists a uniqueconditional copula C W such thatH W (x; yjw) =C W (F W (xjw);G W (yjw)jw) for all (x; y) 2 [¡1; 1] 2 and each w 2 −:Conversely, for any conditional distribution functions F W and G W and any conditionalcopula C W , the function H W de¯ned above is a conditional two-dimensionaldistribution function with marginals F W and G W . ¥There exists a complication, coming from the assumption that the conditionalvariable W must be the same for both marginal distributions and the copula. Itfollows a supporting example.13

Example (a pitfall of the conditional copula, Patton (2006)). Consider XjW 1 ,Y jW 2 and C W1 ;W 2(:j W 1 ;W 2 )andspecifyH W1 ;W 2(x; yjW 1 ;W 2 )=C W1 ;W 2(F W1 (xjW 1 );G W2 (yjW 2 ) j W 1 ;W 2 ):Then, H W1 ;W 2(x; 1j W 1 ;W 2 )=F W1 (xjW 1 ) which is the conditional marginal distributionof (X; Y )jW 1 . By analogy, H W1 ;W 2(1;yjW 1 ;W 2 )=G W2 (yjW 2 ), the conditionaldistribution of (X; Y )jW 2 . Thus the function H W1 ;W 2can not be the joint distributionof (X; Y ) j W 1 ;W 2 in general. The only exception is the very special case whenF W1 (xjW 1 )=F W1 (xjW 1 ;W 2 )andG W2 (yjW 1 )=G W2 (yjW 1 ;W 2 ).The conditional bivariate copula de¯nitioncanbereformulatedintermsofsubalgebraA containing the past information on the process of interest as follows.De¯nition (conditional copula, Fermanian and Scaillet (2005)). For everysub-algebra A the conditional copula with respect to A, associated with a vectorX =(X 1 ;::: ;X n ) is a random function C(:jA) :[0; 1] n ! [0; 1] such thatH(xjA) =C(F X1 (x 1 jA);::: ;F Xn (x n jA) jA)almost surely for every x =(x 1 ;::: ;x n ) 2 [¡1; 1] n . Such a function is uniqueon the product of values taken by the conditional marginal cumulative distributionfunctions F Xi (:jA);i=1;::: ;n:Even if this de¯nition is useful it is di±cult to meet a real situation such that certainvariables a®ect the conditional distribution of one variable but not the other. Inpractice, the marginal distributions are usually de¯ned with respect to past marginalvalues. Consider for example the Markovian process S i;t ; i =1;::: ;n: One workseasily with conditional distribution of S i;t knowing S 0;t , than with the conditional distributionof S i;t knowing the full vector S 0 =(S 0;1 ;::: ;S 0;n ), related with past historyof the process. Actually, it is much simpler to model future returns of a stock index,say SI 1 , knowing the past history of this index (by some state-space or stochasticvolatility models) than knowing the past values of SI 1 and an additional one, SI 2say.The practitioners often prefer to implement their marginal models in informationsystems and the idea is to use them for analysis of more complex multivariate models.The necessity to de¯ne the notion of conditional pseudo-copula is motivated by thefact that marginal processes are usually better known than dependence structures. Inorder to formalize this mathematically, let us consider the sub-algebras A 1 ;::: ;A n ,B and denote A =(A 1 ;::: ;A n ). These sub-algebras can not be chosen arbitrarily,but assuming the following condition: Let x and y be n-dimensional vectors. Foralmost every w 2 − the relation P (X i · x i jA i )(w) =P (X i · y i jA i )(w) for everyi =1;::: ;n implies P (X · xjB)(w) =P (X · yjB)(w). For example, this conditionis satis¯ed when all conditional distributions of X 1 ;::: ;X n are strictly increasing, orwhen A 1 = ¢¢¢= A n = B, as in Patton (2006).14

De¯nition (conditional pseudo-copula, Fermanian and Scaillet (2005)). Theconditional pseudo-copula with respect to the sub-algebras A 1 ;::: ;A n ; A; B and associatedwith X is a random function C pseudo (:jA; B) :[0; 1] n ! [0; 1] such thatH(xjB) =C pseudo (F X1 (x 1 jA 1 );::: ;F Xn (x n jA n )jA; B)almost surely for every x =(x 1 ;::: ;x n ) 2 (¡1; 1) n . Such a function is uniqueon the product of values taken by the conditional marginal cumulative distributionfunctions F Xi (:jA i );i=1;::: ;n:The function C pseudo (:jA; B) is called a pseudo-copula because it satis¯es all propertiesof the usual copula, except the condition 3 in the Formal copula de¯nition.Example (conditional pseudo-copulas are not copulas in general, Fermanianand Wegkamp (2004)). Consider the bivariate processfX n = aX n¡1 + ² n ; Y n = bX n¡1 + cY n¡1 + º n g; n =1; 2;:::;where the sequence of innovation terms ² n and º n are independent Gaussian whitenoises. Set A n;1 = ¾(X n¡1 = x n¡1 ), A n;2 = ¾(Y n¡1 = y n¡1 )andB n = ¾((X; Y ) n¡1 =(x; y) n¡1 ). ThenP (Y n

Sklar's Theorem (for conditional pseudo copula, Fermanian and Wegkamp(2004)). Let H be a joint distribution function on (¡1; 1) n . Assume that for everyx =(x 1 ;::: ;x n ), y =(y 1 ;::: ;y n ) 2 (¡1; 1) n ,F Xi (x i )=F Xi (y i ) for all 1 · j · n implies H(x) =H(y):Then there exist a conditional pseudo-copula C pseudo such thatH(x) =C pseudo (F X1 (x 1 );::: ;F Xn (x n ));for every x =(x 1 ;::: ;x n ) 2 (¡1; 1) n . The function C pseudo is uniquely de¯nedon RanF X1 £ :::£ RanF Xn , the product of the values taken by the F Xi . Conversely,if C pseudo is a conditional pseudo-copula and if F X1 ;::: ;F Xn are some univariatedistribution functions, then the function H is an n-dimensional distribution function.¥Note that the conditional pseudo-copula C pseudo in the last theorem is a copulaif and only if H(+1;::: ;x i ;::: ;1) = F Xi (x i ) for every i = 1;::: ;n and x =(x 1 ;::: ;x n ) 2 (¡1; 1) n .Additional statements and estimation of conditional pseudo-copulas as well asapplications to goodness of ¯t test are discussed by Fermanian and Wegkamp (2004).Fermanian and Scaillet (2005) consider statistical pitfalls arising when using copulas,in particular they discuss issues in copula estimation and the design of time-dependentcopulas. They provide a simulation study where it is shown the potential impact ofmisspeci¯ed margins on the estimation of the copula parameter.2.5 An application: using copula to bound quantile measuresConsider a decision maker faced with a number of risks, i.e. random future losses. Arisk measure à is de¯ned as a mapping from the set of random variables representingthe risks at hand to the real line, i.e., à :(¡1; 1) n ! (¡1; 1). In this section wewill always consider random variables as losses, i.e. random payments that have tobe made.As a ¯rst example, consider the p-quantile risk measure, often called VaR (Valueat-Risk)at level p 2 (0; 1), de¯ned byVaR p (X) =inffx 2 (¡1; 1) :F X (x) ¸ pg = F ¡1X(p):For a given risks (X 1 ;::: ;X n )andariskmeasureà :(¡1; 1) n ! (¡1; 1) oneisofteninterestedincomputing certain quantities of Ã(X 1 ;::: ;X n ) like some momentsor a quantile. In the actuarial and ¯nance literature can be found variety of formsof such a risk measure à corresponding to exotic options, basket derivatives, creditderivatives, operational risk, insurance covers, see Embrechts et al. (2003a), Georgeset al. (2001), Patton (2006). Typical examples of à include:16

(i) à 1 (x 1 ;::: ;x n )=x 1 + :::+ x n ;(ii) à 2 (x 1 ;::: ;x n )= P ni=1 (x i ¡ m) + ,wherem>0anda + = max(a; 0);(iii) à 3 (x 1 ;::: ;x n )=( P ni=1 x i ¡ m) +, m>0.The case (i) can be found in the context of Insurance when one would be interestedin some quantile of the join position X 1 + :::+ X n . Such a case occurs when consideringaggregate claims of an insurance portfolio in a given reference period, or whenobserving discount payments associated to a policy at di®erent future moments oftime, see Kaas et al. (2003). The case (ii) corresponds to analyzing an excess-of-losstreaty in reinsurance where the X i 's could be individual claims or insurance lossesdue to di®erent lines of business; The case (iii) has an interpretation to ¯nancialderivatives (e.g. Asian options) or stop-loss reinsurance, see Dhaene et al. (2002b).In practice, ¯rst we work with the marginals and afterwards with the dependencestructure choosing the underlying (more suitable) copula. Once the dependence structureof the risks is speci¯ed, the calculation of quantities related to distribution functionof Ã(X 1 ;::: ;X n ) becomes a computational issue. In many situations only partialor no information about the copula corresponding to (X 1 ;::: ;X n )isknown. Insuchcases it is useful to evaluate the bounds of the distribution function of Ã(X 1 ;::: ;X n ).Such bounds have been found ¯rstly by Makarov (1981) when n =2andÃ(x 1 ;x 2 )=x 1 + x 2 . Later Frank et al. (1987) use a copula approach extending Makarov's results(except for the optimality of the bounds) to include arbitrary increasing continuousfunctions à as follow. Let C L (u; v) =max(u + v ¡ 1; 0) and z 2 (¡1; 1). Then,P (X + Y · z) ¸ sup C L (F (x);G(y)) = Ã(z);x+y=zà ¡1 (p) =inf fF ¡1 (u)+G ¡1 (v)g; p2 (0; 1)C L (u;v)=pand VaR p (X + Y ) · à ¡1 (p). Denuit et al. (1999) extended these results for the casen ¸ 3, when the marginals are the same.Let us consider the multivariate case. Let (X 1 ;::: ;X n )berandomvariableswithdistributions F X1 ;:::;F Xn and associate copula C. Let à :(¡1; 1) n ! (¡1; 1)be increasing and left continuous in the last argument. Denote by à k the function Ãwith the ¯xed arguments x i1 ;:::;x ik for 1 · i 1

Z¾ C;à (s) =dC(F X1 (x 1 );::: ;F Xn (x n ));fÃ(X 1 ;::: ;X n )·sg½ C;à (s) =inf¡ ¡x 1 ;::: ;x n¡1 2(¡1;1) Cdual F X1 (x 1 );::: ;F Xn¡1 (x n¡1 );F Xn á1n¡1 (s)¢¢ ;where à n¡1(s) ¡1 is the left continuous (generalized) inverse function of à n¡1 (s).Theorem (best possible bounds, Embrechts et al. (2003a)). Under the abovenotations if a copula C associated to (X 1 ;:::;X n ) satis¯es C ¸ C 0 and C dual · C1dualfor some given n-copulas C 0 and C 1 , then¿ C0 ;Ã(s) · ¾ C;à (s) · ½ C1 ;Ã(s):Note that, ¾ C;à (s) =P (Ã(X 1 ;:::;X n ) · s) =¤ à (s). Also, the bounds ¿ C0 ;Ã(s)and ½ C1 ;Ã(s) are themselves distribution functions. Since there is not necessarily aunique copula satisfying the inequalities in the last theorem, it is assumed additionallythat there exists a copula C such that C ¸ C 0 and C dual · C1 dual . The copulas C 0and C 1 represent the partial information available about the dependence structure of(X 1 ;:::;X n ). In Embrechts et al. (2003b) is proposed a methodology in order to ¯ndthe bounds for VaR p (¤ à (s)), recently re¯ned by Embrechts and Puccetti (2006).In most cases, the bounds in last theorem does not have closed form. In general,one has to use numerical approximations as proposed by Williamson and Downs(1990). Their algorithm is based on the discretization of ¿ C0 ;Ã(s), ½ C1 ;Ã(s) andonaduality principle due to Frank et al. (1987).Some problems related to the control and optimization of VaRin a portfolio withnon-normal or non-log-normal returns have taken the researchers to look for anotherquantile measures. Such a measure is called Tail Value-at-Risk at level p, andisde¯ned byZ 1TVaR p (X) = 1 VaR p (X)dq; p 2 (0; 1):1 ¡ p pIt is the arithmetic average of the VaR p (X), from p on. Therefore, TVaRis a measureof risk more suitable than VaR because it takes into account the losses beyond theVaR level. Note that always TVaR p (X) > VaR p (X), TVaR 0 (X) = E(X) andTVaR p (X) is a non-decreasing function of p. The fundamental di®erences betweenVaRand TVaR canbesummarizedasfollows: VaRis the \optimistic" lower boundof the tail loses and the TVaR is the expected value of tail losses. Therefore, TVaRcan be interpreted as a \conservative" risk measure.Rockfellar and Uryasev (2000, 2002) use CV aR instead of TVaR and provide analternative de¯nition of CV aR as a coherent measure (i.e. satisfy the sub-additivity,monotonicity, homogeneity and translation invariance axioms, see Artzner et al.18¥

(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

We may also characterize other statistics which are relevant in reliability, lifemodeling or risk analysis. For example, one could be interested in the range X n:n ¡X 1:n or subranges X r1 :n ¡ X r2 :n for r 1 >r 2 . However, in order to derive explicitformulas, we need the joint distribution of X r1 :n and X r2 :n. In the case of independentand identically distributed random variables, Balakrishnan and Cohen (1991) givemore friendly formulas for the density. Nelsen (2003) found the copula C 1;n of X 1:nand X n:n :C 1;n (u; v) =v ¡ [maxf(1 ¡ u) 1 1n + v n ¡ 1; 0g] n ;see also Schmitz (2004).In the general case, the problem is open. One solution is then to use Monte Carlomethods, as suggested by Georges et al. (2001). A recent study on the degree ofassociation of pairs of ordered random variables is provided by Averous et al. (2005).In Anjos et al. (2005) we give a copula representation of the joint distributionfunction of r-th and s-th order statistics corresponding to X and Y giventhe associated copula C as follows. Consider a bivariate distribution function withcontinuous margins and n independent observations from the population (X; Y ).Let (X 1 ;Y 1 );::: ;(X n ;Y n ); n ¸ 2, be a sample from continuous distribution withcopula C and marginals F and G respectively. Let X r:n and Y s:n be the orderstatistics of the sample, 1 · r; s · n: Since F (x) andG(y) are continuous thepairs f(X 1 ;Y 1 );::: ;(X n ;Y n )g can be transformed into f(U 1 ;V 1 );::: ;(U n ;V n )g byU i = F (X i ) » U(0; 1) and V i = G(Y i ) » U(0; 1). Therefore, we get P (X r:n ·x; Y s:n · y) =P (U r:n · u; V s:n · v); where U r:n and V s:n are r-th and s-th orderstatistics corresponding to n independent observations from (U; V ).The marginal distributions of P (U r:n · u; V s:n · v) are Beta distributed randomvariables, i.e. U r:n » Beta(r; n ¡r +1) and V s:n » Beta(s; n¡s+1). Let ¯¡1¯¡1r;n¡r+1 ands;n¡s+1 be the inverses of these Beta distributions. The copula associated to orderstatistics of the pair (X r:n ;Y s:n ) is the same copula of the pair (U r:n ;V s:n ), i.e.C Xr:n ;Y s:n(w; t) =C Ur:n ;V s:n(w; t) =H Ur:n ;V s:n¡¯¡1r;n¡r+1(w);¯¡1s;n¡s+1(t) ¢ :Under the above notations the copula C Ur:n ;V s:nis given byC Ur:n;V s:n(w; t) =nXj=rnX X n!C ¡¯¡1r;n¡r+1(w);¯¡1s;n¡s+1(t) ¢ mm!(j ¡ m)!(k ¡ m)!(n ¡ j ¡ k + m)!k=sm(4)£ [¯¡1r;n¡r+1 (w) ¡ C(¯¡1r;n¡r+1 (w);¯¡1s;n¡s+1 (t))]j¡m£ [¯¡1s;n¡s+1£ [1 ¡ ¯¡1r;n¡r+1 (w) ¡ ¯¡1(t) ¡ C(¯¡1r;n¡r+1 (w);¯¡1s;n¡s+1 (t))]k¡ms;n¡s+1(t)+C¡¯¡1r;n¡r+1 (w);¯¡1where the third summation is over m 2 [max(0;j+ k ¡ n);min(j; k)]:s;n¡s+1 (t)¢ ] n¡j¡k+m ;20

Let R u = P nj=1 IfU j · ug and R v = P nj=1 IfV j · vg. In Barakat (2001) propertiesof joint distribution (R u ;R v ) are investigated, and as consequences, limitingdistribution results are obtained for the vector (X r:n ;Y s:n )where1· r; s · n. For¯xed r; s ¸ 1, as n !1,thepair(r; s) is called ¯xed rank (or the case of extremeorder statistics). When r; s !1as n !1,(r; s) is called increasing rank. Oneparticular rate of increase of special interest is when r n ! ¸1 and s n ! ¸2 as n !1such that 0 · ¸1;¸2 < 1or¸1 =0;¸2 = 1. Additionally, nine other cases coveringthe possible asymptotic distributions of bivariate order statistics are presented. Weconsider only the increasing rank case, but the method elaborated can be appliedsimilarly for the remaining cases.In Anjos et al. (2005) we extract the asymptotic copula, denoted by C a ,fromthelimiting distribution of (R u ;R v ) and use the corresponding approximation to evaluatethe joint distribution function of order statistics (X r:n ;Y s:n ); as follows.Theorem (weak convergence, Anjos et al. (2005)). Let min ¡ n ¡ r; r ¢ !1and min ¡ n ¡ s; s ¢ !1when n !1. Furthermore, let r n ! ¸1 and s n ! ¸2 asn !1such that 0 · ¸1;¸2 < 1 or ¸1 =0and ¸2 =1.Ifr ¡ nupnu(1 ¡ u)! ¿ 1ands ¡ nvpnv(1 ¡ v)! ¿ 2hold for given u; v 2 (0; 1) and ¯xed constants ¿ 1 and ¿ 2 , and additionally½ Ru ;R v= Corr(R u ;R v ) n !1 ¡¡¡¡¡¡¡¡! ½;thenÃPR u ¡ nupnu(1 ¡ u)·r ¡ nu pnu(1 ¡ u);R v ¡ nvpnv(1 ¡ v)·!p s ¡ nvnv(1 ¡ v)d ¡¡¡¡¡!n !1 © ½ (¿ 1 ;¿ 2 );where © ½ is the joint distribution function of bivariate normal distribution with zeromean vector and correlation coe±cient ½ = pC(u;v)¡uv¥uv(1¡u)(1¡v)Since C(w; t) =H(F ¡1 (w);G ¡1 (t)), the asymptotic copula C a of (R u ;R v )isgivenby C a (w; t) =© ½ (© ¡1 (w); © ¡1 (t)) and the corresponding survival copula has the formC a (w; t) =© ½ (© ¡1 (w); © ¡1 (t)):Under conditions in last theorem H Ur:n;Vs:n (u; v) can be approximated byµµ H Ur:n;Vs:n (u; v) ¼ © ½ ©µ¯r;n¡r+1 ¡1 (u); © ¡1 ¯s;n¡s+1 (v) ;since H Ur:n ;V s:n(u; v) =H Ru ;R v(r; s). Using the well know relation C(w; t) =w +t ¡ 1+C(1 ¡ w; 1 ¡ t) whereC(w; t) is the survival copula, we obtain the following21

asymptotic result using u = F (x) andv = G(y) and relation C Xr:n;Y s:n= C Ur:n;V s:n:H Xr:n;Y s:n(x; y) ¼¯r;n¡r+1¡F (x)¢+ ¯s;n¡s+1¡G(y)¢¡ 1+© ½³© ¡1¡ 1 ¡ ¯r;n¡r+1¡F (x)¢¢; ©¡1 ¡ 1 ¡ ¯s;n¡s+1¡G(y)¢¢´:(5)It is worth to note that the copula of (U r:n ;V s:n ) is di®erent than the copula of(R u ;R v ).Now, we give an example of application of (5).Example. Consider the bivariate normal distribution with zero mean, unit variancesand correlation coe±cient ¡0:5. Let us calculate P (X 9:10 · x; Y 10:10 · y). For x =1:5and y =1:8, we have F (x) =0:93319, G(y) =0:96406 and C ¡ F (x);G(y) ¢ =0:897319.By relation (4), the exact value of P (X 9:10 · x; Y 10:10 · y) isP (X 9:10 · x; Y 10:10 · y) =P (U 9:10 · u; V 10:10 · v)=P (U 9:10 · F (x);V 10:10 · G(y))=10 £ C ¡ F (x);G(y) ¢ 9£ ¡ ¢¤ ¡ ¢ 10G(y) ¡ C F (x);G(y) + C F (x);G(y)=0:5902:For our data we calculate½ ==C ¡ F (x);G(y) ¢ ¡ F (x)G(y)pF (x)G(y)(1 ¡ G(y))(1 ¡ F (x))0:897319 ¡ 0:93319 £ 0:96406p0:93319 £ 0:96406(1 ¡ 0:93319)(1 ¡ 0:96406)= ¡0:0504337and using (5) we obtain³H Xr:n ;Y s:n(x; y) ¼ © ½ © ¡1 ¡¯9;10¡9+1 (F (x)) ¢ ; © ¡1 ¡¯10;10¡10+1 ¢´(G(y))´= © ½³© ¡1 (0:85942); © ¡1 (0:69356)´=0:85942 + 0:69356 ¡ 1+© ½³© ¡1 (1 ¡ 0:85942); © ¡1 (1 ¡ 0:69356)=0:5922:As one can see, the asymptotic copula is easy to calculate and gives a good approximationeven when we use a small sample size.3.2 Copulas with multivariate marginalsSklar's theorem holds whenever the dimension n ¸ 2, so most of the results could beused. But, since it is much more convenient to work in dimension n = 2, practitioners22

usually wish to aggregate two-dimensional framework to obtain a multidimensionalone. Even though the construction of families of joint distribution functions forgiven univariate marginals has been so widely studied, the case of higher dimensionalmarginals has been focused more on study of compatibility of overlapping marginalsand bounds for the corresponding Frechet classes, see Joe (1997), Chapter 3.In dimension n = 3, consider the class H(H X1 X 2;H X1 X 3)with¯xedorknownbivariate marginals H X1 X 2and H X1 X 3, assuming that the ¯rst univariate marginal F X1is the same. This class is always non-empty since it contains trivariate distributionswhich are such that the second and the third variables X 2 and X 3 are conditionallyindependent given the ¯rst one X 1 , i.e. always is possible to ¯nd the joint distributionH(x 1 ;x 2 ;x 3 )=Z x1¡1H X2 jX 1(x 2 jx)H X3 jX 1(x 3 jx)dF X1 (x):Moreover, observe that in such a case the usual Frechet boundsmaxfF X1 (x 1 )+F X2 (x 2 )+F X3 (x 3 ) ¡ 2; 0g and minfF X1 (x 1 );F X2 (x 2 );F X3 (x 3 )gcan be improved, see Joe (1997). The last relation can be extended to the n-variatedistribution, given two di®erent (n ¡ 1)-dimensional margins, containing (n ¡ 2) variablesin common.If we consider the class H(H X1 X 2;H X1 X 3;H X2 X 3) with ¯xed or known bivariatemarginals H X1 X 2, H X2 X 3and H X2 X 3, then compatibility conditions for bivariatemarginals are obtained by considering two of the three margins to be arbitrary, andthe third bivariate margin to have constraints given the other two. The uniquenessand compatibility conditions are discussed by Joe (1997), see also Dall'Aglio (1972)for some necessary conditions.Therefore, one cannot just select a parametric family of functions with the rightboundary properties and expect them to satisfy the rectangle condition (1) of a multivariatejoint distribution function. Generally, a family of multivariate distributionsmust be constructed through methods such as mixtures, stochastic representationsand limits, see Joe (1997), Chapter 4.In fact, the usual copula theory is devoted to the class of n-variate distributionsH(F X1 ;::: ;F Xn ), in which the univariate margins F X1 ;::: ;F Xn are given, see (2).The insu±ciency of the copula function to handle distributions with given multivariatemarginals is illustrated by the Nutshell copula's paradox discussed by Genestet al. (1995). They showed that the only possibility thatH(x 1 ;::: ;x n1 ;x n1 +1;::: ;x n1 +n 2)=C(H n1 (x 1 ;::: ;x n1 );H n2 (x n1 +1;::: ;x n1 +n 2))de¯nes a (n 1 + n 2 )-dimensional distribution function, n 1 + n 2 ¸ 3, for all H n1 andH n2 (with dimensions n 1 and n 2 , respectively) is the independence copula.Schweizer and Sklar (1981) o®ered the following serial iterative approach for constructingcopulas: Consider some bivariate copula C, setC 2 (u 1 ;u 2 )=C(u 1 ;u 2 )anditerateC k (u 1 ;::: ;u k¡1 ;u k )=C(C k¡1 (u 1 ;::: ;u k¡1 );u k ); k ¸ 3;23

ution functions, see Section 2.1.1 and Nelsen et al. (2003), in order to model the dependencestructure between d random vectors (X 1 ;::: ;X m1 ),...,(X md¡1 +1;::: ;X md ),with m d = n using a d-dimensional copula (2 · d · n) to represent the dependencestructure between nonoverlapping marginals which are in general multivariate.The need to study relationships among nonoverlapping random vectors arises naturallyin a variety of circumstances. Such problems arise if one needs to build astochastic model in a situation where the information about the kind of dependenceand knowledge of certain marginal distributions is available. Typically, one has anidea about the dependence mechanism of nonoverlapping segments composing theportfolio, and the basic interest is to search a dependence structure between that segments.Examples are complex engineering systems, Jackson queueing network, controlof risk clustering, pricing and hedging sensitive instruments, pricing and hedgingbasket derivatives and structured products, credit portfolio management, credit andmarket risk measurement, etc. The basic result is the following.Sklar's Theorem (for copulas with multivariate marginals, Anjos and Kolev(2005a)). Let the continuous random variables X 1 ;::: ;X n are distributed into dnonoverlapping random vectors X l =(X ml¡1 +1;::: ;X ml ),withm l = P lj=1 n j, m 0 =0, l =1;::: ;d, 2 · d · n and n = n 1 + :::+ n d .LetH l be the marginal distributionof X l and C Hl be the associated copula of dimension n l , respectively. Then there existsan unique d-dimensional copula C d given byC d (w 1 ;::: ;w d )=P (W 1 · w 1 ;::: ;W d · w d ) (6)for (w 1 ;::: ;w d ) 2 [0; 1] d , representing the dependence structure between X 1 ;::: ;X d ,where W l = K l (H l (X l )) » U(0; 1) and K l is the Kendall distribution function of X l ,l =1;::: ;d. ¥One needs to impose restrictions on marginals H l in order C d to be a n-dimensionaldistribution function of (X 1 ;::: ;X n ). As Marco and Ruiz-Rivas (1992) showed, inthe simplest case, for an arbitrary d-dimensional copula those restrictions lead to maxin¯nitelydivisible marginals. The Nutshell copula's paradox discussed by Genest etal. (1995) is a consequence of such limitations.Problems of this kind arise if one needs to build a stochastic model in a situationwhere the information about the kind of dependence and knowledge of certainmarginal distributions is available. The method based on the theorem above, is motivatedby the following fact. Typically, one has an idea about the dependence mechanismof nonoverlapping segments composing the portfolio, and the basic interest isto search a dependence structure between that segments.The statement of the theorem can be considered as a version of the Sklar's theoremfor copulas with nonoverlapping multivariate marginals. If substitute d = n in (6),one gets (2), i.e. the conclusion of the Sklar's theorem (when the univariate marginalsare ¯xed or known).25

3.3 Copula representation via a local measure of dependenceThe characterization of copulas as well as the choice of the dependence structure aredi±cult problems. For example, the choice of the copula does not inform explicitlyabout the type of the dependence structure between variables involved. As one cansee, the primary task is just to choose an appropriate copula function, where themarginal distributions are treated as nuisance parameters. But what is the meaningof \appropriate"? The aim here is to give a partial answer of this question.The key statistic in a joint Gaussian distribution is the correlation coe±cient. Usually,a low correlation coe±cient between two markets implies a good opportunity foran investor to diversify his investment risk. Hence, based on Gaussian assumption,an investor can signi¯cantly reduce his risk by balancing his portfolio with investmentsintheforeignmarket.Embrechtsetal. (2002) discussed some pitfalls of theusual correlation coe±cient as a global measure of dependence. A Gaussian copularepresents the dependence structure of a joint normal distribution, but a Gaussiancopula does not necessarily imply a joint normal distribution unless the marginals arealso normal. There is increasing evidence indicating that Gaussian assumptions areinappropriate in the real world. It has been found that correlations computed withdi®erent conditions could di®er dramatically, see for example Ang et al. (2002) whostudied the correlations between a portfolio and the market conditional on downsidemovements. It has been found that correlations conditional on large movements arehigher than that conditional on small movements. This phenomena has also beencharacterized as \correlation breakdown", and it is widely discussed in the literature.Boyer et al. (1999) proposed that in situations of \correlation breakdown", thecorrelations can reveal little about the underlying nature of dependence, see also theapproach suggested by Engle (2002). Therefore, although conditional correlationsprovide more information about the dependence than the usual Pearson's correlation,the results are sometimes misleading and need to be interpreted very carefully.In general, the local dependence measures are attractive because they o®er a moreprecise radiography of dependence structure than the corresponding global measures.The local correlation dependence measures are usually de¯ned as a correlation betweenX and Y given X = x and Y = y (or given X · x and Y · y). For example,Kotz and Nadarajah (2002) proposed a new local measure derived from the linearcorrelation coe±cient as followsµ £X ¤£ ¤ E ¡ E(XjY = y) Y ¡ E(Y jX = x)°(x; y) = qE £ X ¡ E(XjY = y) ¤2 E £ Y ¡ E(Y jX = x) ¤ ; (x; y) 2 [¡1; 2 1]2 :The measure °(x; y) is a radical generalization of the usual Pearson's correlationcoe±cient and characterizes the e®ect of X on Y (and vise versa) conditionally on(X; Y ) being at the point (x; y) 2 [¡1; 1] 2 . The \conditional" correlation coe±cient°(x; y) overcomes the weakness of other known measures of local dependence, see for26

example Doksum et al. (1994), Jones (1998), Charpentier (2003) and referencestherein.However, the measure of local dependence °(x; y) is in°uenced by the marginals.This means that for the same dependence structure the change of the marginals wouldimply the change of the measure. From Sklar's Theorem we know that dependencestructure do not depend on marginals. Therefore, the measure °(x; y) is not suitablefor describing the dependence structure in the sense of Sklar's Theorem.In order to avoid this weakness, let us consider a simple modi¯cation based on thefact that the Spearman coe±cient ½ S of (X; Y ) is equal to the Pearson's correlationcoe±cient between F (X) =U and G(Y )=V ,i.e.µ £U ¤£ ¤ E ¡ E(U) V ¡ E(V ) Z 1 Z 1·¸½ S = p =12 C(u; v) ¡ uv dudv;Var(U)Var(V )e.g. Nelsen (1999), p. 138. So, an alternative version of °(x; y) canbegivenbyµ £U ¤£ ¤ E ¡ E(UjV = v) V ¡ E(V jU = u)° S (u; v) = qE £ U ¡ E(UjV = v) ¤ 2 £ ¤ ; (u; v) 2 [0; 1] 2 ;2E V ¡ E(V jU = u)00which is already scale invariant with respect to the uniform marginals. The measure° S can be interpreted as a \conditional" Spearman coe±cient. Nevertheless, we areunable to suggest a copula representation based on ° S .We propose here a local dependence measure di®erent than existing ones, whichcan be used as a new tool for representation of the bivariate copulas. The importanceis that we can decompose any copula C in two parts: the marginals and the dependencestructure embodied in the local dependence measure that can be interpretedas a \local" Spearman coe±cient (but not \conditional" as ° S is). We denote it by½ C and call it Spearman function.In Anjos and Kolev (2005b) is shown that for every bivariate copula C there existsa unique continuous function ½ C such that, for all (u; v) 2 [0; 1] 2 we haveC(u; v) =uv + ½ C (u; v) p uv(1 ¡ u)(1 ¡ v):The advantage is that ½ C provides an explicit and precise information of theunderlying dependence structure. A connection with the Gaussian copula is givenbelow.Theorem (Anjos and Kolev (2005b)). Let C beacopulaandlet© r be a Gaussiancopula with correlation coe±cient r. Then for each pair (u; v) 2 [0; 1] 2 there exist aunique smallest valuer u;v = inffr :© r¡© ¡1 (u); © ¡1 (v) ¢ ¸ C(u; v)g 2[¡1; 1]; (7)27

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

Fisher-Tippett Theorem (version for copulas). Let H 2 MDA(W ) and W (x 1 ;x 2 )=C ¤ (W 1 (x 1 );W 2 (x 2 )). Then(i) The marginals W i are GEV and F Xi 2 MDA(W i ), i =1; 2;(ii) C ¤ (u t 1;u t 2)=C t ¤(u 1 ;u 2 ) for all t>0. ¥It is interesting to note that the limit copula C ¤ is determined only by copulaC associated to H. Thus H 2 MDA(W ) is equivalent to (C; F X1 ;F X2 ) 2MDA(C ¤ ;W 1 ;W 2 ). Then we may writeTheorem. (C; F 1 ;F 2 ) 2 MDA(C ¤ ;W 1 ;W 2 ) if and only if(i) F i 2 MDA(W i ) ;i=1; 2;(ii) C 2 MDA(C ¤ ). ¥Since the copula of the limiting distribution is unique, we need the followingde¯nition.De¯nition (copula domain of attraction). Let C ¤ be an extreme value copula.We say that C is in the domain of attraction of C ¤ , and denote C 2 CDA(C ¤ ), if andonly if C(F X1 ;F X2 ) 2 MDA(C ¤ (W 1 ;W 2 )), for F Xi continuous and F Xi 2 MDA(W i ),i =1; 2.It follows from the max-stability property, see Embrechts et al. (1997), thatC ¤ 2 CDA(C ¤ ), for all extreme value copula C ¤ .Copula modelling of extremes may also be addressed by modelling joint excessesover high thresholds. In this case there are many suitable parametric families available,see Joe (1997). Frees and Valdez (1998) worked out the expression of the copulapertaining to the bivariate Pareto distribution (Clayton copula). Juri and WÄuthrich(2002) characterize the limiting dependence structure in the upper-tails of two randomvariables assuming their dependence structure is Archimedean. Charpentier (2003)studied conditional copulas, i.e. copulas conditional to extreme events and derivedtheir properties. He studied in detail the case of Archimedean copulas and providedapplications in credit risk. Charpentier (2004) focused on the dependence structureof extreme events and compared asymptotic results by considering componentwisemaxima and joint excesses over high thresholds. He took the distributional approachand obtained copula-convergence theorems.Estimation of copulas for extreme values may follow the approaches already mentioned.Results in the literature include the investigation of the behavior of the maximumlikelihood estimators of copula parameters through simulations by Caperaµa etal. (1997) in the case of the symmetric and asymmetric logistic model, and by Genest(1987) in the case of the Frank family. Genest (1987) found that the method ofmoments estimator appears to have smaller mean squared error than the maximumlikelihood estimator at small samples. Hsing et. al. (2004) derive a nonparametricestimation procedure for estimating the limiting copula of componentwise maxima.For statistical tests on copula speci¯cation see Ane and Kharoubi (2003), Genest and30

Rivest (1993), Fermanian and Scaillet (2005).Important applications are found in the ¯elds of Insurance, Environment, Finance.Since marginal ¯ts present no di±culties using extreme value models, one can focuson modelling dependency through a variety of copula families. In Finance, the numberof publications applying copula based techniques is large. The main motivationcomes from the observed greater integration among world markets, which results instronger co-movements, increasing risk and reducing the opportunities for internationaldiversi¯cation. Li (2000) provides a Gaussian copula-based methodology toprice ¯rst-to-default credit derivatives. The issues of measuring ¯nancial risk andassessing the e®ect of copula choice on risk measures were addressed by a number ofauthors, including Embretchs et al. (2003), Ane and Kharoubi (2003), Mendes andSouza (2004), among others. The modelling of daily ¯nancial returns also motivatedMendesandArslan(2005)toobtaintheexpression for the GT-copula. This is thecopula associated to the multivariate GT-distributions which includes as a specialcase the copula pertaining to the multivariate t-distribution. For a survey of ¯nancialapplications see, for example, Bouye et al. (2000) and Embretchs et al. (2003b).E®ective risk management techniques must include extreme value techniques. Inparticular, one non-linear measure of dependence at extreme levels, the tail dependencecoe±cient has become very popular.De¯nition (upper and lower tail dependence coe±cients). Let X 1 and X 2 berandom variables with distribution functions F X1 and F X2 . The coe±cient of uppertail dependence is de¯ned by¸U = lim®!0 + ¸U(®) = lim®!0 + PrfX 1 >F ¡1X 1(1 ¡ ®)jX 2 >F ¡1X 2(1 ¡ ®)g ;provided a limit ¸U 2 [0; 1] exists. If ¸U 2 (0; 1], then X 1 and X 2 are said tobe asymptotically dependent in the upper tail. If ¸U =0,theyareasymptoticallyindependent. Similarly, the lower tail dependence coe±cient is given by¸L = lim®!1 ¡ ¸U(®) = lim®!1 ¡ PrfX 1

et al. (2000). To measure contagion they computed the tail dependence function¸U(®). Malevergne and Sornette (2002) used extreme value theory to obtain closedform expressions for tail dependence generated by factors models. Stocks possessingminimal empirical tail dependence coe±cient with the market were selected to composeportfolios. They found these portfolios possess superior behavior during stressedtimes. Breymann et al. (2003) analyze in detail high frequency data. They establishthe change in dependence as a function of the sampling frequency, and presenta method for deseasonalizing bivariate returns for varying time horizons. Mendes(2005) assessed the extent of integration between stock markets using ¸L and ¸U torank twenty one pairs of emerging stock markets returns during stressful periods.The empirical investigation in Mendes (2005) focused on detecting asymmetriesin data. The methodology consisted of ¯tting positively associated copulas to simultaneousexceedances of high thresholds, testing for independence, and computingcopula based measures of interdependence and contagion. The value of the tail dependencecoe±cients were compared to the Pearson's linear correlation coe±cientand its conditional version °(x; y) noted in Section 3.3. An issue not investigatedin Mendes (2005) deserves close attention. It is the assessment of the e®ect of thetime-varying conditional volatility on marginal and copula ¯ts, in particular on thedependence measures estimates. As a further development, we provide an illustrationusing emerging stock market indexes. An comprehensive empirical investigation of theextent to which interdependence in emerging markets may be driven by conditionallong range dependence in volatility may be found in Mendes and Kolev (2006).Example (assessing the e®ects of FIGARCH volatility on marginal andcopula ¯ts). Our data consist of 2608 observations of daily stock emerging marketindexes returns from January, 3, 1994 to December, 31, 2003. This period includesexamples of extreme market events such as the Mexican devaluation (end of 1994),the Brady bond crisis (beginning of 1995), the Asian series of devaluation (during1997), the Russian crisis (end of 1998), the Brazilian devaluation (end of 1999), theArgentinian crisis (since 2000). Crises in Latin American and East Asian economiesusually result in considerable depreciations of national currencies and have importantglobal repercussions.For the illustration that follows we use the indexes representing the stock marketsof Korea and Singapore. Korea is the second largest emerging market with atotal market capitalization of US$ 298 billion, and Singapore is the ¯fth, with US$148.5 billion. More speci¯cally, the data consists of daily closing levels of the SeoulComposite (Korea) and Singapore Straits Industrial (Singapore) indexes.We start by de¯ning thresholds in each margin and collecting the joint data abovethethresholds. Totheunivariateexcesseswe¯tageneralizationoftheGPD,themodi¯ed GPD (MGPD). The MGPD distribution function is, for some °;¾ > 0,F (y) =1¡ ¡1µ1+» y° »; if » 6= 0;¾32

de¯ned on fy : y>0and(1+» y° ) > 0g. The special case of » = 0, interpreted as¾the limit case » ! 0, gives the constrained modelF (y) =1¡ expµ¡ y°¾; if » =0;for y>0. When °1 induces concavity. The Weibull distribution correspondsto » = 0. Two sub-classes of the Weibull are: the class of sub-exponential distributions,generated by letting °1.The choice of thresholds is a critical issue, not addressed here. After some experimentationobserving the standard errors of marginal parameters estimates, thethresholds were ¯xed as the empirical p i -quantile, (p 1 ;p 2 )=(0:18; 0:17). We note thatthe limit distributional result, the Pickands-Balkema-de Haan theorem, given for theright tail, holds for su±ciently extreme thresholds, but high thresholds may result inlarge variability of estimators (see Embrechts et al., 1997). To estimate the marginaland copula parameters we use the fully parametric approach in two steps, for detailssee Genest et al. (1995) and Joe (1997), based on maximum likelihood estimation.We found the daily marginal (lower tail) excesses follow di®erent MGPD(°;»;¾) distributions,respectively, an MGPD(1:62; 0:56; 6:38) and an MGPD(1:32; 0:33; 1:43).These estimates and respective standard errors are given in Table 2. The ¯rst columngives the threshold values, the second gives the number of joint observationsused (172), and the third, fourth and ¯fth columns show the marginal results, forKorea and Singapore. The marginal excesses are not identically distributed and thusthe excesses are not exchangeable.Table 2: Margins and copula parameters estimates and (standard errors) and ¸U,for the negative co-exceedances of Korea and Singapore.Excess daily returns (raw data)Thresholds # ° » ¾ Copula(parameter) ¸U-1.38 -1.05 172 1.53 1.32 0.56 0.33 6.38 1.43 Gumbel(1.35) 0.33(0.08) (0.05) (0.11) (0.10) (1.32) (0.17) (0.05) (0.14)Excess residuals (¯ltered data)Thresholds # ° » ¾ Copula(parameters) ¸U-0.85 -0.87 147 1.02 1.04 0.13 0.13 0.50 0.56 ALM(1.19, 0.96, 0.82) 0.19(0.06) (0.06) (0.09) (0.08) (0.06) (0.07) (0.07, 0.02, 0.02) (0.09)For the dependence structure we considered four copula families possessing uppertail dependence. They are the Gumbel, see Joe (1997), Joe-Clayton (family BB7 inJoe (1997)), the copula associated to the Kimeldorf and Sampson or Clayton copula,thecopulapertainingtothebivariatePareto distribution, see Frees and Valdez (1998),33

and the non-exchangeable copula (ALM) associated to the asymmetric logistic model,seeTawn(1988)andMendes(2005). UsingtheAICcriterionwefoundthebestcopula¯t as the Gumbel copula with ± =1:35, and tail dependence coe±cient ¸U =0:33.To assess the e®ects of volatility, the univariate data are previously ¯ltered usingthe Fractionally Integrated GARCH (FIGARCH) model. This °exible class ofmodels, able to capture the e®ects of long memory (or fractional integration) in theconditional variance, was introduced by Baillie et al. (1996), and Bollerslev andMikkelsen (1999). To de¯ne a FIGARCH process, let fX t g t2Z be a stochastic processwith zero mean and X t = ¾ t Z t ,whereZ t is an independent identically distributedrandom variable with zero mean and unit variance, such that X t jF t¡1 are independentidentically distributed with mean 0 and variance ¾t 2 ,whereF t¡1 denotes the ¾-¯eldof the information set up to time t ¡ 1. Let º t = Xt 2 ¡ ¾t 2 . The conditional volatility¾ t of a FIGARCH(r; d; s) processisde¯nedford 2 [0; 1] throughwith¾ 2 t = ! (1 ¡ ¯(L)) ¡1 + ¸(L)X 2 t ;¸(L) =1¡ (1 ¡ ¯(L)) ¡1 Á(L)(1 ¡L) d and (1 ¡L) d =1+1X± d;k L k ;¡(k¡d)where ± d;k = ¡d , L is the lag operator, Á = ® + ¯, ®(L) =P r¡(k+1)¡(1¡d) i=1 ® iL i and¯(L) = P sj=1 ¯jL j .Notethatwhend = 0 it reduces to a GARCH model.To model the serial dependence in the mean and variance of the daily returns fromKorea and Singapore we ¯t combinations of ARMA(p; q) andFIGARCH(r; d; s) processes.We estimate by maximum likelihood the models derived by considering allcombinations p; q; r and s in f0; 1; 2g. We choose the best model using the AIC criterion.The best ¯t to Korea and Singapore turned out to be, respectively, ARMA(1; 0)-FIGARCH(1;d;1), and ARMA(1; 0)-FIGARCH(2;d;1) with all parameters estimateshighly signi¯cant, see Table 4. In this table, A stands for the constant in the varianceequation, AR are the autoregressive terms in the model for the mean, ARCH(1) andARCH(2) de¯ne the orders r, andGARCH(1)istheorders.Then, using the same proportions (0:18; 0:17) to de¯ne the thresholds on the ¯lteredseries free of volatility clusters, we obtained the FIGARCH standardized residualsexcesses. The MGPD distributions are ¯tted to the excess data, and the resultsare given in Table 3. Statistical tests applied to the empirical and estimated distributionsof ¯ltered excesses did not reject the hypothesis of identically distributed(p-value= 0:0723). Figure 1 illustrates, and shows the scatter plot of bivariate data,using the (raw) daily excess data (at left), and the ¯ltered excesses (at right). Theformal tests carried on and Figure 1 suggest that non-exchangeability may be a resultof short and long memory in volatility. However, even though these excess residualsare identically distributed, they failed to be exchangeable: We found the best ¯t to bethe one provided by the ALM asymmetric copula, with parameters estimates equalto (1:19; 0:96; 0:82), yielding ¸U =0:19, see Table 2.34k=1

Table 3: ARMA + FIGARCH ¯ts to daily returnsfrom Korea and Singapore indexes.Korea : ARMA(1; 0)+FIGARCH(1;d;1) Singapore: ARMA(1; 0)+FIGARCH(2;d;1)Value Std.Error t value Pr(> jtj) Value Std.Error t value Pr(> jtj)AR(1) 0.11763 0.02084 5.646 9.110e-009 0.1151 0.019923 5.775 4.298e-009A 0.06975 0.02772 2.517 5.954e-003 0.0295 0.008441 3.495 2.410e-004GARCH(1) 0.53477 0.06354 8.416 0.000e+000 0.7562 0.046910 16.121 0.000e+000ARCH(1) 0.22228 0.04286 5.186 1.154e-007 0.3750 0.057193 6.557 3.303e-011ARCH(2) 0.1561 0.029486 5.295 6.453e-008fraction 0.39866 0.05110 7.802 4.441e-015 0.4855 0.043910 11.057 0.000e+000Nelsen (2005a) studies non-exchangeability and proposes a non-exchangeabilitymeasure for the case of identically distributed margins. The degree of exchangeabilitymay be measured by computing the maximum of the absolute value of the di®erencesH(x; y) ¡ H(y; x). For identically distributed margins with joint distribution H andcopula C, the set of values of jH(x; y) ¡ H(y; x)j are the same of jC(u; v) ¡ C(v; u)j.Thus he proposes to compute 3maxjC(u; v) ¡ C(v; u)j, for all u; v 2 [0; 1] 2 .Herewesuggest to use an empirical version of this statistics, using the observed pairs andthe ¯tted ALM copula. Note that, by the Glivenko-Cantelli theorem, as the samplesize goes to in¯nity, the sample version should approach the true value. The valueobtained using the empirical version of Nelsen's measure of non-exchangeability was0:0058.Summarizing, we provided an example where non-exchangeability was found foridentically distributed random variables, and long memory in volatility was responsiblefor changes in dependence structure, increasing extremal dependence. Workscombining copulas and short memory processes (GARCH type) modelling includeGoorbergh et al. (2005) and Breiman et al. (2003) among others.5 ConclusionsThe independence assumptions, which are typical in many statistical models are oftendue more to convenience rather than to the problem in hand. Furthermore, there aresituations where neglecting dependence e®ects may occur into a dramatic underestimationfor quantity of interest (some appropriate risk measure, for example). Takingcare of dependencies becomes therefore important in order to extend standard modelstowards more e±cient ones. However, relaxing the independence assumption yieldsmuch less tractable models. The pitfall of the copula approach is that it is usuallydi±cult to choose or ¯nd the appropriate copula for the problem in hand. An alternativeis suggested in Section 3.3. Often, the only possibility is to start with someguess such a parametric family of copulas and then to try to ¯t the parameters. Asa consequence, the model obtained may su®er a certain degree of arbitrariness.35

0 2 4 6 8 10 120 1 2 3 4 5 60 2 4 6 8 10Daily returns of Korea & Singapore0 1 2 3 4 5 6Standardized residuals of Korea & SingaporeFigure 1: Excesses (left) and standardized residuals excesses (right)Because of space limitations, it was impossible even to comment the di±cultieswith copulas associated to discrete multivariate distributions, the important aspects incopula theory related to stochastic ordering, e.g. MÄuller and Stoyan (2001), graphicaltechniques, e.g. Genest and Boies (2003), classical and new measures of dependence,e.g. Nelsen (1999) and Genest and Plante (2003), etc. A part of these interestingtopics are discussed in Anjos et al. (2004). We would like to mention ¯nally theimportance of survival copulas in life modelling. An excellent overview is given byGeorges et al. (2001), see also Nelsen (2005b).6 AknowledgementsApartofthepresentedmaterialinSection3was,infact,aplanned¯rststepsoftheresearch project related with a visit of the ¯rst author in CIMAT, Guanajuato, Mexicoin the beginning of 2003. As a result, in Kolev et al. (2006) is given an alternativede¯nition of the probability generating function of arbitrary binary vector, wherecomonotonic and countermonotonic discrete copulas play a key role.Special thanks to the organizers of the 8th Symposium of Probability and StochasticProcesses provided in Puebla, Mexico, June 20-25, 2004, where the authors havebeen invited to present their recent results.The authors are thankful to Professor Roger Nelsen for the continuous attentionon our research. We are greatful to the annonimous referee whose critical remarksreduced the errors and improved this overview.The ¯rst and the third authors are partially supported by FAPESP, Grant 03/10105-2 and PROBRAL (CAPES/DAAD), Grant 171-04. The second author thanks for a¯nancial support from CNPq and FAPERJ Grant E-26/170.725/2004.36

ReferencesAlsina,C.,Nelsen,R.B.,Schweizer,B.(1993). On the characterization of a class ofbinary operations on distribution functions. Statistics & Probability Letters 17,85-89.Ane, T., Kharoubi, C. (2003). Dependence structure and risk measure. Journal ofBusiness 76, 411-438.Ang, A., Chen, J., Xing, Y. (2002). Downside correlation and expected stock returns.Working paper No. 01-25, Columbia Business School. Available athttp://ssrn.com/abstract=282986Anjos,U.,Ferreira,F.,Kolev,N.,Mendes,B.(2004). Dependence Modelling viaCopulas. Short course in 16th SINAPE, Caxambu-MG, Brazil. (in Portuguese,English version under preparation). University Press USP: S~ao Paulo.Anjos, U., Kolev, N., Tanaka, N. (2005). Copula associated to order statistics.Brazilian Journal of Probability and Statistics 19, 111-123.Anjos, U., Kolev, N. (2005a). An application of Kendall distribution. Forthcomingin Journal for Economy and Management L(1), 95-101.Anjos, U., Kolev, N. (2005b). Representation of bivariate copulas via local measureof dependence. Technical Report RT-MAE 2005-03, University of S~ao Paulo.Artzner, P., Delbean, F., Eber, J.M., Heath, D. (1999). Coherent measures of risk.Mathematical Finance 9, 203-228.Averous, J., Genest, C., Kochar, S. (2005). On the dependence structure of orderstatistics. Journal of Multivariate Analysis 94, 159-171.Baillie, R.T., Bollerslev, T., Mikkelsen, H.O. (1996). Fractionally integrated generalizedautoregressive conditional heteroskedasticity. Journal of Econometrics74, 3-30.Barakat, H. (2001). The asymptotic distribution theory of bivariate order statistics.Ann. Inst. Stat. Math. 53, 487-497.Balakrishnan, N., Cohen, A.C. (1991). Order Statistics and Inference. AcademicPress: San Diego.Bjerve, S., Doksum, K. (1993). Correlation curves: measures of association as functionsof covariate values. Annals of Statistics 21, 890-902.Bollerslev, T., Mikkelsen, H.O. (1999). Long-term equity anticipation securities andstock market volatility dynamics. Journal of Econometrics 92, 75-99.37

Bouye, E., Durrleman, V. Nikeghbali, A., Riboulet, G., Roncalli, T. (2000). Copulasfor ¯nance: a reading guide and some applications. Working paper. Availableat http://gro.creditlyonnais.fr/content/rd/home copulas.htmBoyer, B.H., Gibson, M.S., Loretan, M. (1999). Pitfalls in tests for changes incorrelations. International Finance Discussion Paper No. 597, Board of FederalReserve System.Breymann, W., Dias, A., Embrechts, P. (2003). Dependence structures for multivariatehigh-frequency data in ¯nance. Quantitative Finance 3, 1-16.Caperaµa, P., Fougµeres, A.L., Genest, C. (1997). A nonparametric estimation procedurefor bivariate extreme value copulas. Biometrika 84, 567-577.Charpentier, A. (2003). Tail distribution and dependence measures. In ProceedingsXXXIV International ASTIN Colloquium, Berlin.Charpentier, A. (2004). Extremes and dependence: a copula approach. In Proceedingsof the 3rd Conference in Actuarial Science & Finance in Samos. Availableat http://www.crest.fr/pageperso/lfa/charpent/charpent.htmChebrian, A., Denuit, M., Scaillet, O. (2002). Testing for concordance ordering. DPFAME 41.Cherubini, U., Luciano, E., Vecchiato, W. (2004).Wiley Finance: Chichester.Copula Methods in Finance.Cohen, L. (1984). Probability distributions with given multivariate marginals. Journalof Mathematical Physics 25, 2402-2403.Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. SpringerSeries in Statistics, Springer: Berlin.Costinot, A., Roncalli, T., Teiletche, J. (2000). Revisiting the dependence between¯nancial markets with copulas. Working paper. Available athttp://gro.creditlyonnais.fr/content/rd/home copulas.htmCuadras, C. (1992). Probability distributions with given multivariate marginals andgiven dependence structure. Journal of Multivariate Analysis 42, 51-66.Dall'Aglio, G. (1972). Frechet classes and compatibility of distribution functions.Symp. Math. 9, 131-150.de Haan, L. (1985). Extremes in higher dimensions: the model and some statistics.Proc. 45th Sess. Int. Statist. Inst., paper 26.3, Amsterdan: InternationalStatistical Institute.38

Darkiewicz, G., Dhaene, J., Goovaerts, M. (2005). Risk measures and dependenciesof risks. Forthcoming in Brazilian Journal of Probability and Statistics 19,155-178.Deheuvels, P. (1978). Caracterisation complµete des lois extr^emes multivariees etla convergence des types extr^emes. Publications de l'Institut de Statistiquedel'Universite deParis23, 1-36.Deheuvels, P. (1979). La funcion de dependance empirique et ses proprietes. Untest non parametrique d'independance. Acad. R. Belg., Bull. Cl. Sci. 5, Serie65, 247-292.Deheuvels, P. (1981a). A Kolmogorov-Smirnov type test for independence and multivariatesamples. Rev. Roum. Math. Pures et Appl., TomeXXVI,2, 213-226.Deheuvels, P. (1981b). A nonparametric test for independence. Publicacions del'ISUP 26, 29-50.Demarta, S. (2001). Extreme Value Theory and Copulas. Master Thesis, Departmentof Mathematics, ETH ZÄurich, Switzerland. Available athttp://www.math.ethz.ch/»demartaDenuit, M., Genest, C., Marceau, E. (1999). Stochastic bounds on sums of dependentrisks. Insurance: Mathematics and Economics 25, 85-104.Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D. (2002a). The conceptof comonotonicity in actuarial science and ¯nance: theory. Insurance:Mathematics and Economics 31, 3-33.Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D. (2002b). The conceptof comonotonicity in actuarial science and ¯nance: applications. Insurance:Mathematics and Economics 31, 133-161.Doksum, K., Blyth, S., Bradlow, E., Meng, X.L., Zhao, H. (1994). Correlationcurves as local measures of variance explained by regression. Journal of theAmerican Statistical Association 89, 571-582.Embrechts, P., KlÄuppelberg, C., Mikosch, T. (1997). Modelling Extremal Events forInsurance and Finance. Springer-Verlag: Berlin.Embrechts,P.,McNeil,A.,Strauman,D.(2002). Correlation and dependence inrisk management: properties and pitfalls. In Risk Management: Value at Riskand Beyond, (Dempster, M., Mo®att, H.K. (Eds.)). Cambridge Univ. Press:Cambridge, 176-223.39

Embrechts, P., Lindskog, F., McNeil, A. (2003a). Modelling Dependence with Copulasand Applications to Risk Management. In Handbook of Hevy Tailed distributionsin Finance, (Rachev, S.T. (Ed.)). Elsvier: Amsterdam, 329-384.Embrechts, P., HÄoing, A., Juri, A. (2003b). Using copulae to bound the Value-at-Risk for functions of dependent risks. Finance & Stochastics 7, 145-167.Embrechts, P., Puccetti, G. (2006). Bounds for functions of multivariate risks.Journal of Multivariate Analysis 97, 526-547.Engle, E. (2002). Dynamic conditional correlation: A Simple class of multivariateGARCH models. Journal of Business and Economic Statistics 20, 339-350.Feller, W. (1968). An Introduction to Probability Theory and Its Applications. VolumeI, 3rd edition. John Wiley & Sons: New York.Fermanian, J-D. (2005). Goodness of ¯t tests for copulas. Journal of MultivariateAnalysis 95, 119-152.Fermanian, J-D., Radulovich, D., Wegkamp, M. (2004). Weak convergence of empiricalcopula process. Bernoulli 10, 847-860.Fermanian J-D., Scaillet, O. (2003). Nonparametric estimation of copulas for timeseries. Journal of Risk 5, 25-54.Fermanian, J-D., Scaillet, O. (2005). Some statistical pitfalls in copula modellingfor ¯nancial applications. In Capital Formation, Governance and Banking (E.Klein, ed.), Nova Science Publishers, 59-74.Fermanian, J-D., Wegkamp, M. (2004). Time dependent copulas. Working paper.Available at http://www.crest.fr/pageperso/lfa/fermanian/fermanian.htmFisher, R.A., Tippet, L.H.C. (1928). Limiting Forms of the Frequency Distributionof the Largest or Smallest Member of a Sample. Proceedings of the CambridgePhilosophical Society 24, 180-190.Frank, M., Nelsen, R., Schweizer, B. (1987). Best possible bounds for the distributionof sum - a problem of Kolmogorov. Probability Theory and Related Fields74, 199-211.Fredricks, G.A., Nelsen, R.B. (1997). Copulas constructed from diagonal sections. InDistributions with Given Marginals and Moment Problems, (Benes, V., Stepan,J. (Eds.)), Kluwer Academic Publishers: Dordrecht, 129-136.Fredricks, G.A., Nelsen, R.B. (2002). The Bertino family of copulas. In Distributionswith Given Marginals and Statistical Modelling, (Cuadras, C.M., Fortiana, J.,Rodrguez Lallena, J.A. (Eds.)), Kluwer Academic Publishers: Dordrecht, 81-91.40

Frees, E.W, Valdez, E. (1998). Understanding Relationships Using Copulas. NorthAmerican Actuarial Journal, 2, 1-25.Galambos,J.(1978).The Asymptotic Theory of Extreme Order Statistics. Wiley &Sons: New York.Genest, C. (1987). Frank's family of bivariate distributions. Biometrika 74, 549-555.Genest, C., Rivest, C. (1993). Statistical inference procedures for bivariate Archimediancopulas. Journal of the American Statistical Association 88, 1034-1043.Genest, C., Ghoudi, K., Rivest, L.P. (1993). A semiparametric estimation procedureof dependence parameters in multivariate families of distributions. Biometrika82, 543-552.Genest, C., Quesada-Molina, J.J., Rodrigues-Lallena, J.A. (1995). De l'impossibilitede construire des lois µa marges multidimensionalles donnees µa partir de copules.C. R. Acad. Sci. Paris 320, 723-726.Genest, C., Quesada-Molina, J.J., Rodriguez Lallena, J.A., Sempi, C. (1999). Acharacterization of quasi-copulas. Journal of Multivariate Analysis 69, 193-205.Genest, C., Plante, J.F. (2003). On Blests measure of rank correlation. The CanadianJournal of Statistics 31, 1-18.Genest, C., Boies, J.-C. (2003). Detecting dependence with Kendall plots. TheAmerican Statistician 57, 275-284.Georges, P., Lamy, A-G., Nicolas, G., Quibel, G., Roncalli, T. (2001). Multivariatesurvival modeling: a uni¯ed approach with copulas. Working paper, Credit Lyonnais.Available at http://gro.creditlyonnais.fr/content/rd/home copulas.htmGhoudi, K., Khoudraji, A., Rivest, L.P. (1998). Proprietes statistiques des copulesde valeurs extr^emes bidimensionneles. Canadian J. of Statistics 26, 187-197.Goncalves,M.,Kolev,N.,Fabris,E.(2005). Bounds for distortion functions ofdependent risks via copulas. In Proc. Second Brazilian Conference on StatisticalModelling in Insurance and Finance, (Eds. N. Kolev and P. Morettin), (2005),122-127.Goorbergh, R.W.J. van der, Genest, C., Werker, B.J.M. (2005). Multivariate optionpricing using dynamic copula models. Forthcoming in Insurance: Mathematicsand Economics.He®ernan, J.E. (2001). A directory of coe±cients of tail dependence. Extremes 3,279-290.41

Hsing, T, KlÄuppelberg, C., Kuhn, G. (2004). Modelling, estimation and visualizationof multivariate dependence for risk management. Technical report, TechnischeUniversitat Minchen. Available at http://www-m4.ma.tum.de/Papers/index.en.htmlHÄurlimann, W. (2002). Analytical bounds for two value-at-risk functionals. ASTINBulletin 32, 235-265.Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall:London.Jones, M.C. (1998). Constant local dependence. Journal of Multivariate Analysis64, 148-155.Juri, A., WÄuthrich, M.V. (2002). Tail dependence from a distributional point ofview. Preprint. Available at http://www.math.ethz.ch/»juri/publications.htmlKaas,R.,Goovaerts,M.,Dhaene,J.,Denuit,M.(2003). Modern Actuarial RiskTheory. 2nd edition, Kluwer Academic Publishers: Dordrecht.Kellerer, H. (1964a). Masstheoretische Marginalprobleme. Math. Ann. 153, 168-198.Kellerer, H. (1964b). Verteilungsfunctionen mit gegebenen Marginalverteilungen.Zeit. Wahrscheinlichkeitsth. 3, 247-270.Kolev, N., Kolkovska, E., Lopez-Mimbela, J.A. (2006). Joint probability generatingfunction for a vector of arbitrary indicator variables. Journal of Computationaland Applied Mathematics 186, 89-98.Kotz, S., Nadarajah, S. (2002). Local dependence functions for the elliptically symmetricdistributions. Sankhy¹a A 65, 207-223.Li, D. (2000). On default correlation: A copula function approach. Journal of FixedIncome 9, 43-54.Li, H., Scarsini, M., Shaked, M. (1996). Linkages: a tool for the construction ofmultivariate distributions with given nonoverlapping multivariate marginals.Journal of Multivariate Analysis 56, 20-41.Li, H., Scarsini, M., Shaked, M. (1999). Dynamic linkages for multivariate distributionswith given nonoverlapping multivariate marginals. Journal of MultivariateAnalysis 68, 54-77.Li, X.P., Mikusinski, V.,Sharewood, H., Taylor, M. (1997). On approximation ofcopulas. In Distributions with Given Marginals and Moment Problems, (Benes,V., Stepan, J. (Eds.)), Kluwer Academic Publishers: Dordrecht, 107-116.42

Longin, F., Solnik, B. (2001). Extreme correlation of international equity markets.Journal of Finance 56, 651-678.Makarov, G. (1981). Estimates for the distribution function of a sum of two randomvariables when the marginal distributions are ¯xed. Theory of Probability andits Applications 26, 803-806.Malevergne, Y., Sornette, D. (2002). Minimizing extremes. Risk 15, 129-133.Marco, J.M., Ruiz-Rivas, C. (1992). On the construction of multivariate distributionswith given nonoverlapping multivariate marginals. Statistics & ProbabilityLetters 15, 259-265.Mendes, B.V.M., Souza, R.M. (2004). Measuring ¯nancial risks with copulas. InternationalReview of Financial Analysis 13, 27-45.Mendes, B.V.M. (2005). Asymmetric extreme interdependence in emerging equitymarkets. Applied Stochastic Models in Business and Industry. 1, 1-16.Mendes, B.V.M., Arslan, O. (2006). Multivariate skew distributions based on theGT-copula. Forthcoming in Brazilian Review of Econometrics.MÄuller, A., Stoyan, D. (2001). Comparison Methods for Stochastic Models and Risks.Wiley: New York.Nadarajah,S.,Mitov,K.,Kotz,S.(2003). Local dependence functions for extremevalue distributions. Journal of Applied Statistics 30, 1081-1100.Nelsen, R. (1995). Copulas, characterization, correlation and counterexamples.Mathematics Magazine 68, 193-198.Mendes, B.V.M., Kolev, N. (2006). Does Long Memory in Volatility Increase Interdependence?Forthcoming in International Review of Financial Analysis,15.Nelsen, R.B., Quesada-Molina, J.J., Schweizer, B., Sempi, C. (1996). Derivabilityof some operations on distribution functions, In Distributions with FixedMarginals and Related Topics, (Ruschendorf, L., Schweizer, B., M. D. Taylor,M.D., (Eds.)), IMS Lecture Notes Monograph Series Number 28: Hayward,233-243.Nelsen, R.B. (1999). An Introduction to Copulas. Springer: New York.Nelsen, R., Quesada-Molina, J., Rodrigues-Lallena, J. and beda-Flores, M. (2001a).Distribution function of copulas: a class of bivariate probability integral transforms.Statistics & Probability Letters 54, 277-282.43

Nelsen, R., Quesada-Molina, J., Rodrigues-Lallena, J. and beda-Flores, M. (2001b).Bounds on bivariate distribution functions with given margins and measures ofassociation. Communications in Statistics: Theory and Methods 30, 1155-1162.Nelsen, R.B., Quesada-Molina, J.J., Rodriguez-Lallena, J.A., Ubeda-Flores, M.(2002a). Multivariate Archimedian quasi-copulas. In Distributions with GivenMarginals and Statistical Modelling, (Cuadras, C.M., Fortiana, J., Rodrguez-Lallena, J.A. (Eds.)), Kluwer Academic Publishers: Dordrecht, 179-185.Nelsen, R.B., Quesada-Molina, J.J., Rodriguez-Lallena, J.A., Ubeda-Flores, M.(2002b). Some new properties of quasi-copulas. In Distributions with GivenMarginals and Statistical Modelling, (Cuadras, C.M., Fortiana, J., Rodrguez-Lallena, J.A. (Eds.)), Kluwer Academic Publishers: Dordrecht, 187-194.Nelsen, R.B. (2003). Properties and applications of copulas: a brief survey. In Proceedingsof the First Brazilian Conference on Statistical Modeling in Insuranceand Finance, (Dhaene, J., Kolev, N., Morettin, P.A. (Eds.)), University PressUSP: S~ao Paulo, 10-28.Nelsen, R.B., Quesada-Molina, J., Rodrigues-Lallena, J., Ubeda-Flores, M. (2003).Kendall distribution functions. Statistics & Probability Letters 65, 263-368.Nelsen, R.B., Quesada-Molina, J.J., Rodriguez-Lallena, J.A., Ubeda-Flores, M.(2004). Best-possible bounds on sets of bivariate distribution functions. Journalof Multivariate Analysis 90, 348-356.Nelsen, R.B. (2005a). Extremes of nonexchangeability. Submitted.Nelsen, R.B. (2005b). Some properties of Schur-constant survival models and theircopulas. Brazilian Journal of Probability and Statistics 19, 179-190.Nelsen, R.B. (2006). An Introduction to Copulas, 2nd Edition, Springer: New York.Niewiadomska-Bugaj, M., Kowalczyk, T. (2005). On grade transformation and itsimplications for copulas. Brazilian Journal of Probability and Statistics 19,125-137.Patton, A. (2006). Modelling asymmetric exchange rate dependence. Forthcomingin International Econometric review.Pickands, J. (1981). Multivariate Extreme Value Distributions. Proceedings of the43rd Session of the I.S.I., 859-878.Reiss, R.D., Thomas, M. (1997). Statistical Analysis of Extreme Values. BirkhÄauser-Verlag: Germany.44

Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes.Springer-Verlag: New York.Rockafellar, R., Uryasev, S. (2000). Optimization of conditional value-at-risk. Journalof Risk 2, 21-41.Rockafellar, R., Uryasev, S. (2002). Conditional value-at-risk for general loss distributions.Journal of Banking & Finance 26, 1443-1471.Rohatgi, V.K. (1984). Statistical Inference. Wiley:NewYork.RÄuschendorf, L. (1985). Construction of multivariate distributions with given marginals.Ann. Inst. Statist. Math. 37 part A, 225-233.Sancetta, A., Satchell, S.E. (2004). Bernstein copula and its applications to modellingand approximations of multivariate distributions. Econometric Theory20, 535-562.Schmitz, V. (2004). Revealing the dependence structure between X (1) and X (n) .Journal of Statistical Planning and Inference 123, 41-47.Schweizer, B., Sklar, A. (1974). Operations on distributions functions not derivablefrom operations on random variables. Studia Math. 52, 43-52.Schweizer, B., Sklar, A. (1981). Probabilistic Metric Spaces. Elsevier: New York.Shi, J., Louis, T. (1995). Inferences on the association parameter in copula modelsfor bivariate survival data. Biometrics 51, 1384-1399.Sklar, A. (1959). Fonctions de repartition a n dimensions et leurs marges. Publ.Inst. Statist. Univ. Paris 8, 229-231.Szczesny, W. (1991). On the performance of a discriminant function. Journal ofClassi¯cation 8, 201-215.Tawn, J. (1988). Bivariate extreme value theory: models and estimation. Biometrika75, 397-415.Wang, S. (2000). A class of distortion operators for pricing ¯nancial and insurancerisks. Journal of Risk and Insurance 67, 15-36.Williamson, R., Downs, T. (1990). Probabilistic arithmetic: numerical methodsfor calculating convolutions and dependency bounds. Journal of ApproximateReasoning 4, 89-158.45