The document discusses unifying standard multivariate copula families that exhibit tail dependence properties. It begins with an overview of common copula families, including elliptical distributions (e.g. normal or spherical), Archimedean copulas, and extreme value distributions. It then discusses the concept of tail dependence, including tail indexes and limiting distributions. The goal is to provide a framework for understanding relationships between different copula families in terms of their tail behavior.

1. 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
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2. 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
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3. 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 F1 , ..., Fd are univariate c.d.f., then
F (x1 , ..., xn ) = C(F1 (x1 ), ..., Fd (xd )) ∀(x1 , ..., xd ) ∈ Rd (1)
is a multivariate c.d.f. with F ∈ F(F1 , ..., Fd ).
2. Conversely, if F ∈ F(F1 , ..., Fd ), there exists a copula C satisfying (1). This copula
is usually not unique, but it is if F1 , ..., Fd are absolutely continuous, and then,
−1 −1
C(u1 , ..., ud ) = F (F1 (u1 ), ..., Fd (ud )), ∀(u1 , , ..., ud ) ∈ [0, 1]d (2)
−1 −1
where quantile functions F1 , ..., Fn are generalized inverse (left cont.) of Fi ’s.
If X ∼ F , then U = (F1 (X1 ), · · · , Fd (Xd )) ∼ C.
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4. 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 (x1 , ..., xn ) = C (F 1 (x1 ), ..., F d (xd )) ∀(x1 , ..., xd ) ∈ Rd (3)
is a multivariate s.d.f. with F ∈ F(F1 , ..., Fd ).
2. Conversely, if F ∈ F(F1 , ..., Fd ), there exists a copula C satisfying (3). This
copula is usually not unique, but it is if F1 , ..., Fd are absolutely continuous, and
then,
−1 −1
C (u1 , ..., ud ) = F (F 1 (u1 ), ..., F d (ud )), ∀(u1 , , ..., ud ) ∈ [0, 1]d (4)
−1 −1
where quantile functions F1 , ..., Fn are generalized inverse (left cont.) of Fi ’s.
If X ∼ F , then U = (F1 (X1 ), · · · , Fd (Xd )) ∼ C and 1 − U ∼ C .
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5. Arthur CHARPENTIER - Unifying copula families and tail dependence
Benchmark copulas
Definition 2
The independent copula C ⊥ is defined as
d
C ⊥ (u1 , ..., un ) = u1 × · · · × ud = ui .
i=1
Definition 3
The comonotonic copula C + (the Fréchet-Hoeffding upper bound of the set of copulas)
is the copuladefined as C + (u1 , ..., ud ) = min{u1 , ..., ud }.
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6. Arthur CHARPENTIER - Unifying copula families and tail dependence
Spherical distributions
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Definition 4
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Random vector X as a spherical distribution if
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X =R·U
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where R is a positive random variable and U is uniformly dis-
tributed on the unit sphere of Rd , with R ⊥ U .
⊥
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E.g. X ∼ N (0, I).
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Those distribution can be non-symmetric, see Hartman & Wintner (AJM, 1940)
or Cambanis, Huang & Simons (JMVA, 1979))
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7. Arthur CHARPENTIER - Unifying copula families and tail dependence
Elliptical distributions
Definition 5
2
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Random vector X as a elliptical distribution if
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X =µ+R·A·U
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where R is a positive random variable and U is uniformly dis-
tributed on the unit sphere of Rd , with R ⊥ U . , and where A
⊥
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satisfies AA = Σ. q
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E.g. X ∼ N (µ, Σ).
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Elliptical distribution are popular in finance, see e.g. Jondeau, Poon & Rockinger
(FMPM, 2008)
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8. Arthur CHARPENTIER - Unifying copula families and tail dependence
Archimedean copula
Definition 6
If d ≥ 2, an Archimedean generator is a function φ : [0, 1] → [0, ∞) such that φ−1 is
d-completely monotone (i.e. ψ is d-completely monotone if ψ is continuous and
∀k = 0, 1, ..., d, (−1)k dk ψ(t)/dtk ≥ 0).
Definition 7
Copula C is an Archimedean copula is, for some generator φ,
C(u1 , ..., ud ) = φ−1 (φ(u1 ) + ... + φ(ud )), ∀u1 , ..., ud ∈ [0, 1].
Exemple1
φ(t) = − log(t) yields the independent copula C ⊥ .
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