Modelling Dependence with Copulas - IFOR

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5 Archimedean Copulas Example 5.2. Let ϕ(t) =(t −θ − 1)/θ, whereθ ∈ [−1, ∞)\{0}. This gives the Clayton family C θ (u, v) =max([u −θ + v −θ − 1] −1/θ , 0). (5.2.6) For θ>0 the copulas are strict and the copula expression simplifies to C θ (u, v) =(u −θ + v −θ − 1) −1/θ . (5.2.7) The Clayton family has lower tail dependence for θ ≥ 0 and is comprehensive, i.e., C −1 = W , C ∞ = M and lim θ→0 C θ = Π. Since most of the following results are results for strict Archimedean copulas we will refer to (5.2.7) as the Clayton family and refer to (5.2.6) as the extension to negative dependence of the Clayton family. Example 5.3. Let ϕ(t) =− ln e−θt −1,whereθ ∈ R\{0}. This gives the Frank e θ −1 family C θ (u, v) =− 1 ( θ ln 1+ (e−θu − 1)(e −θv ) − 1) e −θ . − 1 The Frank copulas are strict copulas and are comprehensive, i.e., C −∞ = W , C ∞ = M and lim θ→0 C θ = Π. Members of the Frank family are the only Archimedean copulas which satisfy the equation C(u, v) =Ĉ(u, v) for so called radial symmetry. Example 5.4. Let ϕ(t) =1−t for t in [0, 1]. Then ϕ [−1] (t) =1−t for t in [0, 1] and 0 for t>1; i.e., ϕ [−1] (t) =max(1−t, 0). Hence C(u, v) =max(u+v−1, 0) = W (u, v). Hence W is Archimedean. The results in the following theorem will enable multivariate extensions of Archimedean copulas. Theorem 5.2. Let C be an Archimedean copula with generator ϕ. Then: 1. C is symmetric; i.e., C(u, v) =C(v, u) for all u, v in I; 2. C is associative; i.e., C(C(u, v),w)=C(u, C(v, w)) for all u, v, w in I. Proof. The first part follows directly from (5.2.2). C(C(u, v),w) = ϕ [−1] (ϕ(C(u, v)) + ϕ(w)) = ϕ [−1] (ϕ(ϕ [−1] (ϕ(u)+ϕ(v))) + ϕ(w)) = ϕ [−1] (ϕ(u)+ϕ(v)+ϕ(w)) = ϕ [−1] (ϕ(u)+ϕ(ϕ [−1] (ϕ(v)+ϕ(w)))) = ϕ [−1] (ϕ(u)+ϕ(C(v, w))) = C(u, C(v, w)), and hence C is associative. The associativity property of Archimedean copulas is not shared by copulas in general as indicated by the following example. Example 5.5. Let C θ be a member of the bivariate Farlie-Gumbel-Morgenstern family of copulas, i.e., C θ (u, v) =uv + θuv(1 − u)(1 − v), for θ ∈ [−1, 1]. Then ( ( 1 1 C θ 4 ,C θ 2 , 1 ( ( 1 ≠ C θ C θ 3)) 4 , 1 ) , 1 ) 2 3 for all θ ∈ [−1, 1] except 0. Hence the only member of the bivariate Farlie-Gumbel- Morgenstern family of copulas that is Archimedean is Π. 32
5.3 Properties 5.3 Properties For convenience, we let Ω denote the set of continuous strictly decreasing convex functions ϕ from I to [0, 1] with ϕ(1) = 0. Theorem 5.3. Let C be an Archimedean copula generated by ϕ in Ω. Let K C (t) denote the C-measure of the set {(u, v) ∈ I 2 |C(u, v) ≤ t}. Then for any t in I, K C (t) =t − ϕ(t) ϕ ′ (t + ) . (5.3.1) Proof. Let t be in (0, 1), and set w = ϕ(t). Let n be a fixed positive integer, and consider the partition of the interval [t, 1] induced by the partition {0,w/n,... ,kw/n,... ,w} of [0,w], i.e., the partition {t = t 0 ,t 1 ,... ,t k ,... ,t n =1} where t n−k = ϕ [−1] (kw/n),k = 0, 1,... ,n. Since w
Page 1: Eidgenössische Technische Hochschu
Page 5: Contents Contents 1 Motivation 1 2
Page 8 and 9: 1 Motivation distributions. Hence w
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: 5.2 Definitions (since ϕ and ϕ [
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 • • •
Page 59 and 60: 7 Extreme Value Copulas 7.1 Univari
Page 61 and 62: 7.2 Multivariate Extreme Value Theo
Page 63: 7.2 Multivariate Extreme Value Theo
Page 66 and 67: 8 Mixture of Extremal Distributions
Page 74 and 75: 9 Modelling Extremal Events in Prac
Page 85: 10 Conclusions In this paper we hav
Page 88 and 89:
11 Appendix { if(rcor[row,col] == 1
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11 Appendix lognormal
Page 93:
References References [1] Dhaene J.
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