Arthur CHARPENTIER - Archimax copulas (and other copula families) Archimax Copulas Arthur Charpentier charpentier.arthur@uqam.ca http ://freakonometrics.hypotheses.org/ based on joint work with A.-L. Fougères , C. Genest and J. Nešlehová March 2014, CIMAT, Guanajuato, Mexico. 1 ✶✶♣t ✶✶♣t ◆♦t❡ ❊①❡♠♣❧❡ ❊①❡♠♣❧❡ ✶✶♣t Pr❡✉✈❡
Arthur CHARPENTIER - Archimax copulas (and other copula families) 2
Arthur CHARPENTIER - Archimax copulas (and other copula families) Agenda ◦ Copulas • Standard copula families ◦ Elliptical distributions (and copulas) ◦ Archimedean copulas ◦ Extreme value distributions (and copulas) • Archimax copulas ◦ Archimax copulas in dimension 2 ◦ Archimax copulas in dimension d ≥ 3 3
Arthur CHARPENTIER - Archimax copulas (and other copula families) Copulas, in dimension d = 2 Definition 1 A copula in dimension 2 is a c.d.f on [0 , 1] 2 , with margins U ([0 , 1]) . Thus, let C ( u, v ) = P ( U ≤ u, V ≤ v ), where 0 ≤ u, v ≤ 1, then • C (0 , x ) = C ( x, 0) = 0 ∀ x ∈ [0 , 1] , • C (1 , x ) = C ( x, 1) = x ∀ x ∈ [0 , 1] , • and some increasingness property 4
Arthur CHARPENTIER - Archimax copulas (and other copula families) Copulas, in dimension d = 2 Definition 2 A copula in dimension 2 is a c.d.f on [0 , 1] 2 , with margins U ([0 , 1]) . Thus, let C ( u, v ) = P ( U ≤ u, V ≤ v ), ● where 0 ≤ u, v ≤ 1, then ● • C (0 , x ) = C ( x, 0) = 0 ∀ x ∈ [0 , 1] , ● • C (1 , x ) = C ( x, 1) = x ∀ x ∈ [0 , 1] , ● • If 0 ≤ u 1 ≤ u 2 ≤ 1, 0 ≤ v 1 ≤ v 2 ≤ 1 ● ● C ( u 2 , v 2 )+ C ( u 1 , v 1 ) ≥ C ( u 1 , v 2 )+ C ( u 2 , v 1 ) ● ● (concept of 2-increasing function in R 2 ) � v � u see C ( u, v ) = c ( x, y ) d x d y with the density notation. � �� � 0 0 ≥ 0 5
Arthur CHARPENTIER - Archimax copulas (and other copula families) Copulas, in dimension d ≥ 2 The concept of d -increasing function simply means that P ( a 1 ≤ U 1 ≤ b 1 , ..., a d ≤ U d ≤ b d ) = P ( U ∈ [ a , b ]) ≥ 0 where U = ( U 1 , ..., U d ) ∼ C for all a ≤ b (where a i ≤ bi ). Definition 3 Function h : R d → R is d -increasing if for all rectangle [ a , b ] ⊂ R d , V h ([ a , b ]) ≥ 0 , where V h ([ a , b ]) = ∆ b a h ( t ) = ∆ b d a d ∆ b d − 1 a d − 1 ... ∆ b 2 a 2 ∆ b 1 a 1 h ( t ) (1) and for all t , with ∆ b i a i h ( t ) = h ( t 1 , ..., t i − 1 , b i , t i +1 , ..., t n ) − h ( t 1 , ..., t i − 1 , a i , t i +1 , ..., t n ) . (2) 6
Arthur CHARPENTIER - Archimax copulas (and other copula families) Copulas, in dimension d ≥ 2 Definition 4 A copula in dimension d is a c.d.f on [0 , 1] d , with margins U ([0 , 1]) . Theorem 1 1. If C is a copula, and F 1 , ..., F d are univariate c.d.f., then F ( x 1 , ..., x n ) = C ( F 1 ( x 1 ) , ..., F d ( x d )) ∀ ( x 1 , ..., x d ) ∈ R d (3) is a multivariate c.d.f. with F ∈ F ( F 1 , ..., F d ) . 2. Conversely, if F ∈ F ( F 1 , ..., F d ) , there exists a copula C satisfying (3). This copula is usually not unique, but it is if F 1 , ..., F d are absolutely continuous, and then, C ( u 1 , ..., u d ) = F ( F − 1 ( u 1 ) , ..., F − 1 d ( u d )) , ∀ ( u 1 , , ..., u d ) ∈ [0 , 1] d (4) 1 where quantile functions F − 1 , ..., F − 1 are generalized inverse (left cont.) of F i ’s. 1 n If X ∼ F , then U = ( F 1 ( X 1 ) , · · · , F d ( X d )) ∼ C . 7
Arthur CHARPENTIER - Archimax copulas (and other copula families) Survival (or dual) copulas Theorem 2 1. If C ⋆ is a copula, and F 1 , ..., F d are univariate s.d.f., then F ( x 1 , ..., x n ) = C ⋆ ( F 1 ( x 1 ) , ..., F d ( x d )) ∀ ( x 1 , ..., x d ) ∈ R d (5) is a multivariate s.d.f. with F ∈ F ( F 1 , ..., F d ) . 2. Conversely, if F ∈ F ( F 1 , ..., F d ) , there exists a copula C ⋆ satisfying (5). This copula is usually not unique, but it is if F 1 , ..., F d are absolutely continuous, and then, − 1 − 1 C ⋆ ( u 1 , ..., u d ) = F ( F d ( u d )) , ∀ ( u 1 , , ..., u d ) ∈ [0 , 1] d 1 ( u 1 ) , ..., F (6) where quantile functions F − 1 , ..., F − 1 are generalized inverse (left cont.) of F i ’s. 1 n If X ∼ F , then U = ( F 1 ( X 1 ) , · · · , F d ( X d )) ∼ C and 1 − U ∼ C ⋆ . 8
Arthur CHARPENTIER - Archimax copulas (and other copula families) Benchmark copulas Definition 5 The independent copula C ⊥ is defined as d � C ⊥ ( u 1 , ..., u n ) = u 1 × · · · × u d = u i . i =1 Definition 6 The comonotonic copula C + (the Fréchet-Hoeffding upper bound of the set of copulas) is the copula defined as C + ( u 1 , ..., u d ) = min { u 1 , ..., u d } . 9
Arthur CHARPENTIER - Archimax copulas (and other copula families) Spherical distributions ● ● ● 2 ● ● ● ● Definition 7 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Random vector X as a spherical distribution if ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● X = R · U ● ● −2 ● ● ● ● ● ● −2 −1 0 1 2 where R is a positive random variable and U is uniformly dis- tributed on the unit sphere of R d , with R ⊥ ⊥ U . ● ● ● 2 0.02 ● ● ● ● 0.04 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.08 ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● . 1 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 . 1 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● E.g. X ∼ N ( 0 , I ). ● ● ● ● ● ● ● ● 0 . 0 6 ● ● ● ● ● ● −2 ● ● ● ● ● ● ● −2 −1 0 1 2 Those distribution can be non-symmetric, see Hartman & Wintner (AJM, 1940) or Cambanis, Huang & Simons (JMVA, 1979)) 10
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