Copulas: An Introduction Part II: Models - …

1 Copulas: An Introduction part II: Models Johan Segers Universit catholique de Louvain (BE). Institut de statistique, biostatistique et sciences actuarielles Columbia University, New York City 9 11 Oct 2013. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 1 / 65. Copulas: An Introduction part II: Models Archimedean copulas Extreme-value copulas Elliptical copulas Vines Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 2 / 65. Copulas: An Introduction part II: Models Archimedean copulas Extreme-value copulas Elliptical copulas Vines Johan Segers (UCL) Copulas.

2 II - Models Columbia University, Oct 2013 3 / 65. The (in)famous Archimedean copulas I By far the most popular (theory & practice) class of copulas I Plenty of parametric Models I Gumbel, Clayton, Frank, Joe, Ali Mikhail Haq, .. I Building block for more complicated constructions: I Nested/Hierarchical Archimedean copulas I Vine copulas I Archimax copulas I .. I Mindless application of (Archimedean) copulas has drawn many criticisms on the copula hype'. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 4 / 65. Laplace transform of a positive random variable Recall the Laplace transform of a random variable Z > 0: Z.

3 (s) = E[exp( sZ)] = e sz dFZ (z), s [0, ]. 0. A distribution on (0, ) is identified by its Laplace transform. Ex. Show the following properties: I 0 (s) 1. I (0) = 1 and ( ) = 0. I ( 1)k dk (s)/dsk 0 for all integer k 1. I In particular, is nonincreasing (k = 1) and convex (k = 2). Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 5 / 65. Survival functions in proportional hazards model : The Laplace transform of the frailty appears Independent unit exponential random variables Y1 , .. , Yd . Survival times X1.

4 , Xd are affected by a common frailty' Z > 0: Xj = Yj /Z. Marginal and joint survival functions: Pr[Xj > xj ] = E[e xj Z ]. = (xj ). Pr[X1 > x1 , .. , Xd > xd ] = E[e (x1 + +xd )Z ]. = (x1 + + xd ). Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 6 / 65. In proportional hazards Models , survival copulas are Archimedean The survival copula of X is Archimedean with generator : 1 , .. , ud ) = 1 (u1 ) + + 1 (ud ).. C(u Ex. Show the above formula. Ex. Show that replacing Z by Z for a constant > 0 changes but does not change the copula.

5 Ex. Pick your favourite (discrete/continuous) distribution on (0, ), compute or look up its Laplace transform, and compute the associated Archimedean copula. If it doesn't exist yet, name it after yourself and publish a paper about it. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 7 / 65. A Gamma frailty induces the Clayton copula If Z Gamma(1/ , 1), with 0 < < , then . z1/ 1 e z Z. (s) = e sz dz = (1 + s) 1/ . 0 (1/ ). and the resulting survival copula is Clayton: . C(u) = (u . 1 + + ud d + 1). 1/ . Ex. Check the above formulas.

6 Ex. How to use the frailty representation to sample from a Clayton copula? Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 8 / 65. Generator of the Clayton copula Generator Inverse generator w = (s) s = 1(w). w s s w w = (s) = (1 + s) 1/ s = 1 (w) = w 1. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 9 / 65. Formal definition of an Archimedean copula A copula C is Archimedean if there exists : [0, ] [0, 1] such that C(u) = 1 (u1 ) + + 1 (ud ).. For C to be a copula, it is sufficient and necessary that satisfies I (0) = 1 and ( ) = 0.

7 I is d-monotone, I ( 1)k dk (s)/dsk 0 for k {0, .. , d 2}. I ( 1)d 2 dd 2 (s)/dsd 2 is decreasing and convex Equivalently, there should exists a random variable Z > 0 such that d 1 . sZ. (s) = E 1 . d 1 +. is the Williamson d-transform of the rv (d 1)/Z. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 10 / 65. Standard examples Ex. The independence copula (u) = u1 ud is Archimedean. I What is its generator ? I What is the frailty variable Z? Ex. The Fr chet Hoeffding lower bound W(u, v) = max(u + v 1, 0) is Archimedean too.

8 What is its generator ? [This is not a Laplace transform; it is 2-monotone but not d-monotone for d 3.]. Ex. One can show that the Fr chet Hoeffding upper bound M(u) = min(u1 , .. , ud ) is not Archimedean. Still, show that the Clayton copula with converges to M. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 11 / 65. Common generator functions w = (s) w = (s). w w s s (u) = u1 ud W(u, v) = max(u + v 1, 0). (s) = e s (s) = max(1 s, 0). Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 12 / 65.

9 Bivariate Archimedean copulas as binary operators A bivariate Archimedean copula induces a binary operator [0, 1] [0, 1] [0, 1] : (u, v) 7 C(u, v). which is commutative and associative: C(u, v) = C(v, u), C(u, C(v, w)) = C(C(u, v), w). endowing [0, 1] with a semi-group structure. Link with the theory of associative functions (A BEL, H ILBERT). Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 13 / 65. Derived quantities Conditional cdf: 0 1 (u1 ) + + 1 (ud ).. C j (u) = . 0 1 (uj ). Pdf, provided is d times continuously differentiable (d) 1 (u1 ) + + 1 (ud ).

10 C(u) = Qd . 0 1. j=1 (uj ). Ex. Show these formulas. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 14 / 65. Yet another probability integral transform: Kendall distribution functions Bivariate cdf H, continuous margins F and G, copula C. The Kendall distribution of a random pair (X, Y) H is the cdf of the rv W = H(X, Y) = C(F(X), G(Y)) = C(U, V). It only depends on H through C: Z. KC (w) = Pr(W w) = 1{C(u, v) w} dC(u, v), w [0, 1]. [0,1]2. It is linked to Kendall's tau via Z 1 Z. 1+ . E[W] = w dKC (w) = C(u, v) dC(u, v) =.