Copulas: a Review and Recent Developments (2007)

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