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Multivariate Copulas

Multivariate Copulas. Bob Fountain May 3, 2005. Grounded. Definition 4.1 The function G on I n is grounded if. n -increasing. Definition 4.2 G is n -increasing if the G -volume of A is non-negative for every n -box A whose vertices lie in the domain of G. Theorem 4.1

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Multivariate Copulas

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  1. Multivariate Copulas Bob Fountain May 3, 2005

  2. Grounded Definition 4.1 The function G on In is grounded if

  3. n-increasing Definition 4.2 G is n-increasing if the G-volume of A is non-negative for every n-box A whose vertices lie in the domain of G.

  4. Theorem 4.1 If G is grounded and n-increasing, then G is non-decreasing in each argument.

  5. k-dimensional margins

  6. 1-dimensional margins

  7. n-dimensional copulas Definition 4.5 A function C on In is a copula if • C is grounded • Each 1-dimensional margin is the identity function • C is n-increasing

  8. Theorem 4.2 The k-dimensional margins of an n-dimensional copula are k-dimensional copulas.

  9. Fréchet Bounds Theorem 4.3 Note: for n>2, the lower bound does not satisfy the definition of a copula.

  10. Sklar’s Theorem Theorem 4.5 Let be marginal distribution functions. (i) If C is a copula, then is a joint distribution function with margins

  11. (ii) If F is a joint distribution function with continuous margins Then there exists a unique copula C such that

  12. Invariance Theorem 4.6 n-dimensional copulas are invariant with respect to increasing transformations.

  13. Increasing/Decreasing Corollary 4.3 If g1 is a.s. decreasing and g2, …, gn are a.s. increasing, then

  14. Joint survival function

  15. Joint survival function (uniform) Definition 4.6

  16. Survival Copula Definition 4.8 The survival copula is the copula such that where

  17. Relationships So and

  18. Theorem 4.8 Let be random variables with continuous c.d.f.s and copula C. Also let be continuous c.d.f.s.

  19. And let Then the margins of the random vector are and the copula is

  20. Density of a copula Definition 4.9 The density associated with a copula is

  21. Multivariate dependence Definition 4.10 Let be an n-dimensional random vector. Then X is positively lower orthant dependent (PLOD) if

  22. Also, X is positively upper orthant dependent (PUOD) if

  23. Multivariate Gaussian copula where is the standardized multivariate normal distribution with correlation matrix R.

  24. Multivariate Student’s t copula where is the standardized multivariate Student’s t distribution with correlation matrix R.

  25. Archimedean copulas A strict generator is which is continuous, strictly decreasing, convex, and

  26. Definition 4.14 An n-variate Archimedean copula is Theorem 4.10 guarantees that this is a copula, provided that is completely monotonic on

  27. Gumbel n-copula

  28. Clayton n-copula

  29. Frank n-copula

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