Modelling Dependence with Copulas - IFOR

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1 Motivation distributions. Hence when modelling dependence, knowledge of copulas is essential. In this paper we discuss a large number of copula families and suggest efficient simulation techniques. The results provide a basis for those interested in modelling dependence in theory and practice. 2
2 <strong>Copulas</strong> 2.1 Mathematical Introduction Definition 1. Let S 1 , S 2 ,...S n be nonempty subsets of R, whereR denotes the extended real line [−∞, ∞]. Let H be a real function of n variables such that DomH = S 1 × S 2 × ...× S n and let B =[a, b] beann-box all of whose vertices are in DomH. Then the H-volume of B is given by V H (B) = ∑ sgn(c)H(c), (2.1.1) where the sum is taken over all vertices c of B, and sgn(c) isgivenby sgn(c) = { 1, if ck = a k foranevennumberofk’s, −1, if c k = a k for an odd number of k’s. (2.1.2) Equivalently, the H-volume of an n-box B =[a, b] isthenth order difference of H on B V H (B) =△ b aH(t) =△bn a n △ bn−1 a n−1 ...△ b1 a 1 H(t), wherewedefinethen first order differences as △ b k ak H(t) =H(t 1 ,... ,t k−1 ,b k ,t k+1 ,... ,t n ) − H(t 1 ,... ,t k−1 ,a k ,t k+1 ,... ,t n ). Definition 2. A real function H of n variables is n-increasing if V H (B) ≥ 0 for all n-boxes B whose vertices lies in DomH. Suppose that the domain of a real function H of n variables is given by DomH = S 1 × S 2 × ...× S n where each S k has a least element a k .WesaythatH is grounded if H(t) = 0 for all t in DomH such that t k = a k for at least one k. If each S k is nonempty and has a greatest element b k ,thenwesaythatH has margins, and the one-dimensional margins of H are the functions H k given by DomH k = S k and H k (x) =H(b 1 ,... ,b k−1 ,x,b k+1 ,... ,b n ) for all x in S k . (2.1.3) Higher dimensional margins are defined by fixing fewer places in H. One-dimensional margins will be called simply “margins” and for k ≥ 2 k-dimensional margins will be called “k-margins”. Lemma 2.1. Let S 1 ,S 2 ,... ,S n be nonempty subsets of R, andletH be a grounded n-increasing function <strong>with</strong> domain S 1 × S 2 × ...× S n . Then H is nondecreasing in each argument, that is, if (t 1 ,... ,t k−1 ,x,t k+1 ,... ,t n ) and (t 1 ,... ,t k−1 ,y,t k+1 ,... ,t n ) are in DomH and x
Page 1: Eidgenössische Technische Hochschu
Page 5: Contents Contents 1 Motivation 1 2
Page 10 and 11: 2 Copulas 1. DomC ′ = S 1 × S 2
Page 12 and 13: 2 Copulas Theorem 2.3. If C ′ is
Page 14 and 15: 2 Copulas Here C α1(X 1),α 2(X 2)
Page 16 and 17: 2 Copulas . • Simulate a value u
Page 18 and 19: 3 Dependence 3.2 Perfect Dependence
Page 20 and 21: 3 Dependence Thus ∫∫ P[(X 1 −
Page 22 and 23: 3 Dependence Proof. For both Kendal
Page 24 and 25: 3 Dependence Definition 11. Let X a
Page 26 and 27: 3 Dependence Furthermore if C is a
Page 28 and 29: 4 Techniques for Construction of Mu
Page 35 and 36: 5 Archimedean Copulas In this chapt
Page 37 and 38: 5.2 Definitions (since ϕ and ϕ [
Page 39 and 40: 5.3 Properties 5.3 Properties For c
Page 41 and 42: 5.3 Properties Example 5.8. Conside
Page 43 and 44: 5.4 Order Theorem 5.7. Let ϕ be a
Page 45 and 46: 5.5 Multivariate Archimedean Copula
Page 47 and 48: 5.5 Multivariate Archimedean Copula
Page 49: 5.5 Multivariate Archimedean Copula
Page 52 and 53: 6 Elliptical Copulas Example 6.2. F
Page 54 and 55: 6 Elliptical Copulas We now address
Page 56 and 57: 6 Elliptical Copulas • • •
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7 Extreme Value Copulas 7.1 Univari
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7.2 Multivariate Extreme Value Theo
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7.2 Multivariate Extreme Value Theo
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8 Mixture of Extremal Distributions
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9 Modelling Extremal Events in Prac
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10 Conclusions In this paper we hav
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11 Appendix { if(rcor[row,col] == 1
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11 Appendix lognormal
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References References [1] Dhaene J.
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