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
q
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Definition 4
q q q
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Random vector X as a spherical distribution if
q qq q
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q qqq q q
q q q q q
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q q qq
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X =R·U
q
q q
−2
q q
q
q q
−2 −1 0 1 2
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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q
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Random vector X as a elliptical distribution if
q
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q
X =µ+R·A·U
q
q q
q q
q
q
q
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−2
−2 −1 0 1 2
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).
q
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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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