By Roger B. Nelsen
The examine of copulas and their function in data is a brand new yet vigorously growing to be box. during this publication the coed or practitioner of records and likelihood will locate discussions of the basic homes of copulas and a few in their fundamental purposes. The functions comprise the learn of dependence and measures of organization, and the development of households of bivariate distributions. This e-book is acceptable as a textual content or for self-study.
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Extra info for An Introduction to Copulas (Springer Series in Statistics)
5; see (Schweizer and Sklar 1983) for details. 4. 10 Multivariate Copulas 45 S1¥ S2 ¥L¥ Sn . Let x = ( x1 , x 2 ,L , x n ) and y = ( y1 , y 2 ,L , y n ) be any points in S1¥ S2 ¥L¥ Sn . Then n H ( x) - H ( y) £ Â H k ( xk ) - H k ( yk ) . k =1 We are now in a position to define n-dimensional subcopulas and copulas. 2. 5. An n-dimensional subcopula (or n-subcopula) is a function C ¢ with the following properties: 1. Dom C ¢ = S1¥ S2 ¥L¥ Sn , where each Sk is a subset of I containing 0 and 1; 2. C ¢ is grounded and n-increasing; 3.
3) We shall encounter this family again in Chapter 4. ■ Two other functions closely related to copulas—and survival copulas—are the dual of a copula and the co-copula (Schweizer and Sklar 1983). The dual of a copula C is the function C˜ defined by C˜ ( u , v ) = u + v - C ( u , v ) ; and the co-copula is the function C * defined by C * ( u , v ) = 1 - C (1 - u ,1 - v ) . Neither of these is a copula, but when C 34 2 Definitions and Basic Properties is the copula of a pair of random variables X and Y, the dual of the copula and the co-copula each express a probability of an event involving X and Y.
If F1 , F2 ,…, Fn are all continuous, C is unique. Otherwise, C is uniquely determined on Ran F1 ¥Ran F2 ¥L¥Ran Fn . 8) n W ( u) = max( u1 + u 2 ,+ L + u n - n + 1,0). 36). 5). 4. 12. If C ¢ is any n-subcopula, then for every u in Dom C ¢ , W n ( u) £ C ¢( u) £ M n ( u) . 13. For any n ≥ 3 and any u in In , there exists an ncopula C (which depends on u) such that C ( u) = W n ( u) . Proof (Sklar 1998). Let u = ( u1 , u 2 ,L , u n ) be a (fixed) point in In other than 0 = (0,0,…,0) or 1 = (1,1,…,1).