Copulas: a Review and Recent Developments (2007)

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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
Page 3 and 4: can be employed in probability theo
Page 5 and 6: y Genest et al. (1993) and Shi and
Page 7 and 8: Table 1: Distribution of (C 1 jC 2
Page 9 and 10: It should be mentioned here that gr
Page 11 and 12: Nelsen et al. (2003) have used Bert
Page 13 and 14: 2.4 Time dependent copulasIn practi
Page 15 and 16: De¯nition (conditional pseudo-copu
Page 17 and 18: (i) Ã 1 (x 1 ;::: ;x n )=x 1 + :::
Page 19 and 20: (1999)) even in the cases where the
Page 21: Let R u = P nj=1 IfU j · ug and R
Page 26 and 27: 3.3 Copula representation via a loc
Page 28 and 29: such thatC(u; v) =© ru;v¡© ¡1 (
Page 30 and 31: Fisher-Tippett Theorem (version for
Page 32 and 33: et al. (2000). To measure contagion
Page 34 and 35: and the non-exchangeable copula (AL
Page 36 and 37: 0 2 4 6 8 10 120 1 2 3 4 5 60 2 4 6
Page 38 and 39: Bouye, E., Durrleman, V. Nikeghbali
Page 40 and 41: Embrechts, P., Lindskog, F., McNeil
Page 42 and 43: Hsing, T, KlÄuppelberg, C., Kuhn,
Page 44 and 45: Nelsen, R., Quesada-Molina, J., Rod

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