Unifying standard multivariate copulas families (with tail - PowerPoint PPT Presentation

Arthur CHARPENTIER - Unifying copula families and tail dependence Unifying standard multivariate copulas families (with tail dependence properties) Arthur Charpentier charpentier.arthur@uqam.ca http ://freakonometrics.hypotheses.org/ inspired by some joint work (and discussion) with A.-L. Fougères , C. Genest , J. Nešlehová , J. Segers January 2013, H.E.C. Lausanne 1 ✶✶♣t ✶✶♣t ◆♦t❡ ❊①❡♠♣❧❡ ❊①❡♠♣❧❡ ✶✶♣t Pr❡✉✈❡

Arthur CHARPENTIER - Unifying copula families and tail dependence Agenda • Standard copula families ◦ Elliptical distributions (and copulas) ◦ Archimedean copulas ◦ Extreme value distributions (and copulas) • Tail dependence ◦ Tail indexes ◦ Limiting distributions ◦ Other properties of tail behavior 2

Arthur CHARPENTIER - Unifying copula families and tail dependence Copulas Definition 1 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 (1) 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 (1). 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 (2) 1 where quantile functions F − 1 , ..., F − 1 are generalized inverse (left cont.) of F i ’s. n 1 If X ∼ F , then U = ( F 1 ( X 1 ) , · · · , F d ( X d )) ∼ C . 3

Arthur CHARPENTIER - Unifying copula families and tail dependence 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 (3) 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 (3). 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 (4) where quantile functions F − 1 , ..., F − 1 are generalized inverse (left cont.) of F i ’s. n 1 If X ∼ F , then U = ( F 1 ( X 1 ) , · · · , F d ( X d )) ∼ C and 1 − U ∼ C ⋆ . 4

Arthur CHARPENTIER - Unifying copula families and tail dependence Benchmark copulas Definition 2 The independent copula C ⊥ is defined as d � C ⊥ ( u 1 , ..., u n ) = u 1 × · · · × u d = u i . i =1 Definition 3 The comonotonic copula C + (the Fréchet-Hoeffding upper bound of the set of copulas) is the copuladefined as C + ( u 1 , ..., u d ) = min { u 1 , ..., u d } . 5

Arthur CHARPENTIER - Unifying copula families and tail dependence Spherical distributions ● ● ● 2 ● ● ● ● Definition 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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)) 6