Distributionworkshop 2.pptx

A RTHUR CHARPENTIER - MULTIVARIATE DISTRIBUTIONS
Multivariate Distributions: A brief overview
(Spherical/Elliptical Distributions, Distributions on the Simplex & Copulas)
A. Charpentier (Université de Rennes 1 & UQàM)
Université de Rennes 1 Workshop, November 2015.
http://freakonometrics.hypotheses.org
1
@freakonometrics

Geometry in Rd and Statistics
The standard inner product is < x, y > A2 = xT y =
Σ
i xi yi.
Hence, x ⊥ y if < x, y > A2 = 0.
A2
The Euclidean norm is ǁxǁ =
1
2
< x, x > =
Σ n
A2 i=1 xi
1
2 2
.
The unit sphere of Rd is S d = { x ∈ Rd : ǁxǁA2 = 1} .
If x = {x1, ···, xn }, note that the empirical covariance is
Cov(x, y) =< x − x, y − y > A2
and Var(x) = ǁx − xǁA2 .
For the (multivariate) linear model, i 0
T
1 i i
y = β + β x + ε , or equivalently,
yi = β0+ < β1, xi > A2 +εi
2
@freakonometrics

The d dimensional Gaussian Random Vector
If Z ∼ N (0, I), then X = AZ + µ ∼ N (µ, Σ ) where Σ = AAT
.
Conversely (Cholesky decomposition), if X ∼ N (µ, Σ), then X = LZ + µ for
T 1
2
some lower triangular matrix L satisfying Σ = LL . Denote L = Σ .
With Cholesky decomposition, we have the particular case (with a Gaussian
distribution) of Rosenblatt (1952)’s chain,
f (x1, x2, ···, xd) = f 1(x1) ·f 2|1(x2|x1) ·f 3|2,1(x3|x2, x1) ···
···f d|d−1,·
·
·,2,1(xd|xd−1, ···, x2, x1).
f (x;µ, Σ) =
1
d
2
(2π) |Σ|
1
2
1
exp − ( T −1
x − µ) Σ (
2
` ˛¸ x
ǁ x ǁ µ , Σ
x − µ) for all d
x ∈R .
3
@freakonometrics

The d dimensional Gaussian Random Vector
Note that ǁxǁµ,Σ = (x − µ)TΣ−1
(x − µ) is the Mahalanobis distance.
Define the ellipsoid Eµ,Σ = { x ∈ Rd : ǁxǁµ,Σ = 1}
Let
X =
X1
∼ N
µ1
,
Σ11 Σ12
X2 µ2 Σ21 Σ22
then
X |
1 2 2
X = x ∼ N (µ + Σ −1
1 22 2
12 2 11 12
−1
22
Σ (x − µ ) , Σ − Σ Σ Σ 21)
X1 ⊥ X2 if and only if Σ12 = 0.
Further, if X ∼ N (µ, Σ), then AX + b∼ N (Aµ + b,AΣAT
).
4
@freakonometrics

The Gaussian Distribution, as a Spherical Distributions
If X ∼ N (0, I), then X = R ·U, where
R 2 = ǁX ǁA2 ∼ χ 2(d)
and
A2 d
U = X/ǁXǁ ∼ 𝐶(S ),
with R ⊥ U.
− 2
− 1
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5
@freakonometrics

The Gaussian Distribution, as an Elliptical Distributions
1
2
` ˛¸ x
If X ∼ N (µ, Σ ), then X = µ + R ·Σ ·U, where
A2
R 2 = ǁX ǁ ∼ χ 2(d)
and
U = X/ǁXǁ A2 d
∼ 𝐶(S ),
with R ⊥ U.
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6
@freakonometrics

Spherical Distributions
Let M denote an orthogonal matrix, M T
M = M M T
= I. X has a spherical
distribution if X =L
M X .
E.g. in R2,
cos(θ) − sin(θ) X 1
=
L X 1
sin(θ) cos(θ) X2 X2
For every a ∈ Rd, aT X =
L
ǁaǁA2 Yi for any i ∈ { 1, ···, d} .
Further, the generating function of X can be written
T
it X T 2
A2
E[e ] = ϕ(t t) = ϕ(ǁtǁ ), ∀ d
t ∈R ,
for some ϕ : R+ → R+.
7
@freakonometrics

Uniform Distribution on the Sphere
Actually, more complex that it seems...
1
x
3
x = ρ sin ϕ cosθ
x2 = ρ sin ϕ sin θ
= ρ cosϕ
with ρ > 0, ϕ ∈[0, 2π] and θ ∈ [0, π].
If Φ ∼ 𝐶([0, 2π]) and Θ ∼ 𝐶([0, π]),
we do not have a uniform distribution on the sphere...
see https://en.wikibooks.org/wiki/Mathematica/Uniform_Spherical_Distribution,
http://freakonometrics.hypotheses.org/10355
8
@freakonometrics

Spherical Distributions
Random vector X as a spherical distribution if
X = R ·U
where R is a positive random variable and U is uniformly
d
d
distributed on the unit sphere of R , S , with R ⊥ U
E.g. X ∼ N (0, I).
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9
@freakonometrics

Elliptical Distributions
Random vector X as a elliptical distribution if
X = µ + R ·A ·U
d
where A satisfies AA'
= Σ, U(S ), with R ⊥ U. Denote
1
Σ 2 = A.
E.g. X ∼ N µ,
( Σ ).
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0.14
10
@freakonometrics

1
2
X = µ + R Σ U where R is a positive random variable, U ∼ 𝐶( d
S ), with
U ⊥ R . If X ∼ F R , then X ∼ E(µ, Σ , F R ).
Remark Instead of F R it is more common to use ϕ such that
E[ei tT
X ] = ei tT
µϕ(tT Σt), t ∈Rd.
E[X] = µ and Var[X] = −2ϕ'(0)Σ
f (x) 𝖺
1
|Σ|
1
2
q
T −1
f ( (x − µ) Σ (x − µ))
where f : R+ → R+ is called radial density. Note that
dF (r) 𝖺 rd−1f (r)1(x > 0).
11
@freakonometrics

If X ∼ E(µ, Σ , F R ), then
AX + b ∼ E(Aµ + b, AΣ AT
, F R )
If
X 2 µ2
X =
X 1
∼ E
µ1
,
Σ11
Σ 21 Σ22
Σ 12
, F R
then
X |
1 2
X = x2 1
∼ E(µ + Σ 12Σ −1
22 (x2 2 11 12
−1
22
− µ ) Σ − Σ Σ Σ 21 1|2
, F )
where
2 1
2
F1|2 is the c.d.f. of (R − *) given X2 = x2.
12
@freakonometrics

Mixtures of Normal Distributions
Let Z ∼ N (0, I). Let W denote a positive random variable, Z ⊥ W . Set
√ 1
2
X = µ + W Σ Z ,
so that X |W = w ∼ N (µ, wΣ ).
E[X ] = µ and Var[X ] = E[W ]Σ
T 1
2
T T
i t X i t µ− W t Σ t) d
E[e ] = E e , t ∈R .
i.e. X ∼ E(µ, Σ, ϕ) where ϕ is the generating function of W , i.e. ϕ(t) = E[e−tW ].
If W has an inverse Gamma distribution, W ∼ IG(ν/2, ν/2), then X has a
multivariate t distribution, with ν degrees of freedom.
13
@freakonometrics

Multivariate Student t
X ∼ t(µ, Σ , ν) if
1 Z
2
X = µ + Σ √
W /ν
where Z ∼ N (0, I) and W ∼ χ 2(ν), with Z ⊥ W .
Note that
Var[X] =
ν
ν − 2
Σ if ν > 2.
14
@freakonometrics

Multivariate Student t
(r = 0.1, ν = 4), (r = 0.9,ν = 4), (r = 0.5, ν = 4) and (r = 0.5, ν = 10).
15
@freakonometrics

On Conditional Independence, de Finetti & Hewitt
1
Instead of X =L
M X for any orthogonal matrix M , consider the equality for any
permutation matrix M , i.e.
(X1, ···, Xd ) =L
(Xσ(1), ···, Xσ(d)) for any permutation of {1, ···, d}
E.g. X ∼ N (0,Σ) with Σi,i = 1 and Σi , j = ρ when i /= j. Note that necessarily
ρ = Corr(X i , X j ) = −
d − 1
.
From de Finetti (1931), X 1, ···, X d, ··· are exchangeable { 0, 1} variables if and
only if there is a c.d.f. Π on [0,1] such that
P[X = x] =
∫
0
θ [1 − θ] dΠ(θ),
1
xT 1 n −xT 1
i.e. X1 , ···, Xd , ··· are (conditionnaly) independent given Θ ∼ Π.
16
@freakonometrics

On Conditional Independence, de Finetti & Hewitt-Savage
More generally, from Hewitt & Savage (1955) random variables X1 , ···, Xd , ···
are exchangeable if and only if there is F such that X1 , ···, Xd , ··· are
(conditionnaly) independent given F.
E.g. popular shared frailty models. Consider lifetimes T1, ···, Td, with Cox-type
proportional hazard µi (t) = Θ ·µi,0(t), so that
i
P[T > t|Θ = θ] = F
θ
i,0(t)
Assume that lifetimes are (conditionnaly) independent given Θ.
17
@freakonometrics

The Simplex Sd ⊂ Rd
(
Sd = x = (x1, x2, ···, x d
d .
.
) ∈ R xi
d
Σ
i =1
)
> 0,i = 1, 2, ···, d; xi = 1 .
Henre, the simplex here is the set of d-dimensional probability vectors. Note that
Sd = { x ∈ Rd
+ : ǁxǁA1 = 1}
Remark Sometimes the simplex is
˜
Sd−1 =
(
x = (x1, x2, ···, xd−1) ∈ Rd−1 .
.
.
i
x > 0, i = 1 2
, , ···, d;
d−1
Σ
i =1
1
)
xi≤ .
Note that if x̃ ∈ S
˜d−1, then (x̃, 1 − x̃T 1) ∈ Sd.
If h : R+
d → R+ is homogeneous of order 1, i.e. h(λx) = λ ·h(x) for all λ > 0.
Then
A1
h(x) = ǁxǁ ·h
x
ǁxǁA1
where
x
ǁxǁA1
d
∈ S .
18
@freakonometrics

Compositional Data and Geometry of the Simplex
C[x1, x2, ···, xd] = Σ d
x
1 2
d
i =1 i =1 x
i i
, , . . . ,
x x xd
Σ Σ d
i =1 xi
Following Aitchison (1986), given x ∈Rd
+ define the closure operator C
" #
d
∈ S .
It is possible to define (Aitchison) inner product on Sd
a
< x, y > =
x y
i i
Σ Σ
i , j i
xi
log log = log log
yi
1
2d x j yj x y
where x denotes the geometric mean of x.
It is then possible to define a linear model with compositional covariates,
yi = β0+ < β1, xi > a +εi .
19
@freakonometrics

Dirichlet Distributions
Given α ∈ Rd
+, and x ∈ Sd ⊂ Rd
1
f (x , ··· d
, x ;α) =
1
B(α)
d
i =1
i
xα i −1
,
where
B(α) =
d
i=1 i
Γ(α )
Γ
Σ d
i =1 αi
Then
Σ d
j =1
j i
α − α .
X i ∼ Beta αi,
and E(X) = C(α).
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20
@freakonometrics

Dirichlet Distributions
Stochastic Representation
Let Z = (Z , ···, Z ) denote independent
1 d i
G(α , θ) random
1 d
T
variables. Then S = Z + ···+ Z = Z 1 has a G( T
α 1, θ)
X = C(X) =
Z
S
= Σ d
i =1 Z i
Z Z
1 d
, ···, Σ d
i =1 Z i
distribution, and
!
has a Dirichlet distribution Dirichlet(α).
0
1
2
3
4
0
1
2
3
4
5 5
0
2
3
5
4
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1
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0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.01.0
0.2
0.4
0.6
0.8
1.0 ●
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21
@freakonometrics

Uniform Distribution on the Simplex
X ∼ D(1) is a random vector uniformly distributed on the simplex.
Consider d − 1 independent random variables U1, ···, Ud−1 with a 𝐶([0,1])
distribution. Define spacings, as
X i = U(i −1):d − U where Ui :d are order
statistics with conventions U0:d = 0and Ud:d = 1. Then
X = (X 1, ···, X d) ∼ 𝐶(Sd).
22
@freakonometrics

‘Normal distribution on the Simplex’
(also called logistic-normal).
Let Y
˜ ∼ N (µ, Σ) in dimension d − 1. Set Z = (Y˜, 0) and
Z
X = C(e ) =
eZ 1 eZ d
eZ 1 + ···+ eZ d
, ···,
eZ 1 + ···+ eZ d
23
@freakonometrics

Distribution on Rd or [0, 1]d
Technically, things are more simple when X = (X1, ···, Xd ) take values in a
product measurable space, e.g. R × ···× R.
In that case, X has independent components if (and only if)
d
P[X ∈ A] = P[X i 1 d
∈ A i ], where A = A × ···, ×A .
i=1
E.g. if A i = (−∞, xi ), then
F (x) = P[X ∈ (−∞ , x] =
d
i =1
i i
P[X ∈ (−∞, x ] =
d
i =1
i i
F (x ).
If F is absolutely continous,
f (x) =
d
∂ F (x)
∂x1 ···∂xd
d
i =1
i i
= f (x ).
24
@freakonometrics

Fréchet classes
Given some (univariate) cumulative distribution functions F1, ···, Fd R → [0,1],
let F(F1, ···, Fd) denote the set of multivariate cumulative distribution function
of random vectors X such that X i ∼ Fi .
Note that for any F ∈ F (F 1, ···, F d), ∀
x ∈ Rd,
F − (x) ≤ F (x) ≤ F +(x)
where
F +(x) = min{Fi(xi), i = 1, ···, d},
and
F −(x) = max{0, F1(x1) + ···+ Fd(xd) − (d − 1)}.
Note that F + ∈ F(F1, ···, Fd), while usually F − ∈
/F(F1, ···, Fd).
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Copulas in Dimension 2
A copula C : [0, 1]2 → [0, 1] is a cumulative distribution function with uniform
margins on [0, 1].
Equivalently, a copula C : [0, 1]2 → [0, 1] is a function satisfying
• C(u1, 0) = C(0, u2) = 0for any u1, u2 ∈[0,1],
• C(u1, 1) = u1 et C(1, u2) = u2 for any u1, u2 ∈ [0,1],
• C is a 2-increasing function, i.e. for all 0 ≤ ui ≤ vi ≤ 1,
C(v1, v2) − C(v1, u2) − C(u1, v2) + C(u1, u2) ≥ 0.
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Borders of the copula function
!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
!0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
! 0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.
Border conditions, in dimension d = 2, C(u1, 0) = C(0, u2) = 0, C(u1, 1) = u1 et
C (1, u2) = u2.
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If C is the copula of random vector (X1, X2), then C couples marginal
distributions, in the sense that
P(X 1 ≤ x1, X 2 ≤ x2) = C (P(X 1 ≤ x1),P(X 2 ≤ x2))
Note tht is is also possible to couple survival distributions: there exists a copula
C ٨ such that
P(X > x, Y > y) = C٨ (P(X > x), P(Y > y)).
Observe that
C٨(u1, u2) = u1 + u2 − 1 + C(1 − u1, 1 − u2).
The survival copula C ٨ associated to C is the copula defined by
C٨(u1, u2) = u1 + u2 − 1 + C(1 − u1, 1 − u2).
Note that (1 − U1, 1 − U2) ∼ C ٨ if (U1, U2) ∼ C .
28
@freakonometrics

If X has distribution F ∈ F(F1, F2), with absolutely continuous margins, then
its copula is
C (u1, u2) = F (F 1
−1
(u1), F 2
−1
(u2)), ∀
u1, u2 ∈ [0, 1].
More generally, if h−1 denotes the generalized inverse of some increasing function
h : R → R, defined as h−1(t) = inf{ x, h(x) ≥ t, t ∈ R} , then
C(u1, u2) = F (F1
−1
(u1), F2
−1
(u2)) is one copula of X.
Note that copulas are continuous functions; actually they are Lipschitz: for all
0≤ ui, vi ≤ 1,
|C (u1, u2) − C (v1, v2)| ≤ |u1 − v1|+ |u2 − v2|.
29
@freakonometrics

Copulas in Dimension d
The increasing property of the copula function is related to the property that
P(X ∈ [a,b]) = P(a1 ≤ X1 ≤ b1,···, ad ≤ X d ≤ bd) ≥ 0
for X = (X1, ···, Xd ) ∼ F , for any a ≤ b(in the sense that ai ≤ bi.
Function h : Rd → R is said to be d-increaasing if for any [a,b]⊂ Rd,
Vh ([a, b]) ≥ 0, where
h
V ([a, b b
a
]) = ∆ h ( bd
ad
t) = ∆ ∆ bd − 1
a d − 1
...∆ 2
b b1
a2 a 1
∆ h (t)
for any t, where
∆ bi
a i
h (t) = h ( 1
t , ···, t , b , t
i −1 i i +1 n
, ···, t ) − h ( 1
t , ···, t , a , t
i −1 i i +1, ··· n
, t ) .
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Black dot, + sign, white dot, - sign.
31
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A copula in dimension d is a cumulative distribution function on [0, 1]d with
uniform margins, on [0,1].
Equivalently, copulas are functions C : [0,1]d → [0,1] such that for all 0≤ ui ≤ 1,
with i = 1, ···, d,
C (1, ···, 1, ui , 1, ···, 1) = ui ,
C (u1, ···, ui −1, 0, ui +1, ···, ud) = 0,
C is d-increasing.
The most important result is Sklar’s theorem, from Sklar (1959).
32
@freakonometrics

Sklar’s Theorem
1. If C is a copula, and if F1 ···, Fd are (univariate) distribution functions,
then, for any (x1, ···, xd) ∈ Rd,
F (x1, ···, xn ) = C (F 1(x1), ···, F d(xd))
is a cumulative distribution function of the Fréchet class F(F1, ···, Fd).
2. Conversely, if F ∈ F(F1, ···, Fd), there exists a copula C such that the
equation above holds. This function is not unique, but it is if margins
F1, ···, Fd are absolutely continousand then, for any (u1, ···, ud) ∈ [0,1]d,
1
−1
1
C(u1, ···, ud) = F (F (u ), ···, F −1
d d
(u )),
where F 1
−1
, ···, F n
−1 are generalized quantiles.
33
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Let (X1, ···, Xd ) be a random vector with copula C. Let φ1, ···, φd, φi : R → R
denote continuous functions strictly increasing, then C is also a copula of
(φ1(X 1), ···, φd(X d)).
If C is a copula, then function
d
C٨(u1, ···, ud) = (−1)k
Σ Σ
k =0 i 1 ,·
·
·,ik
k
C (1, ···, 1, 1 − ui 1 , 1, ...1, 1 − ui , 1, ...., 1) ,
for all (u1, ···, ud) ∈ [0,1] × ... × [0,1], is a copula, called survival copula,
associated with C.
If (U1, ···, Ud) ∼ C , then (1 − U1 ···, 1 − Ud) ∼ C ٨. And if
P(X1 ≤ x1, ···, X d ≤ xd) = C(P(X1 ≤ x1), ···, P(Xd ≤ xd)),
for all (x1, ···, xd) ∈R, then
P(X1 > x1, ···, X d > xd) = C٨(P(X1 > x1), ···, P(Xd > xd)).
34
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On Quasi-Copulas
Function Q : [0,1]d → [0,1] is a quasi-copula if for any 0≤ ui ≤ 1, i = 1, ···, d,
Q(1, ···, 1, ui , 1, ···, 1) = ui ,
Q(u1, ···, ui −1, 0, ui +1, ···, ud) = 0,
s ›→Q(u1, ···, ui−1, s, ui+1, ···, ud) is an increasing function for any i, and
|Q(u1, ···, ud) − Q(v1, ···, vd)| ≤ |u1 − v1| + ···+ |ud − vd|.
For instance, C − is usually not a copula, but it is a quasi-copula.
Let Cbe a set of copula function and define C− and C+ as lower and upper
bounds for C, in the sense that
C−(u) = inf{C(u), C ∈ C} and C+(u) = sup{C(u), C ∈ C}.
Then C− and C+ are quasi copulas (see connexions with the definition of
Choquet capacities as lower bounds of probability measures).
35
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The Indepedent Copula C⊥ , or Π
The independent copula C ⊥ is the copula defined as
⊥
1 n
C (u , ···, u ) =
d
u1 ···ud = ui (= Π(u1, ···, un )).
i=1
Let X ∈ F(F1, ···, Fd), then X⊥
∈ F(F1, ···, Fd) will denote a random vector
with copula C⊥ , called ‘independent version’ of X.
36
@freakonometrics

Fréchet-Hoeffding bounds C − and C+ , and Comonotonicity
Recall that the family of copula functions is bounded: for any copula C,
C−(u1, ···, ud) ≤ C(u1, ···, ud) ≤ C+(u1, ···, ud),
for any (u1, ···, ud) ∈ [0, 1] × ... × [0, 1], where
C−(u1, ···, ud) = max{0, u1 + ... + ud − (d − 1)}
and
C +(u1, ···, ud) = min{ u1, ···, ud} .
If C + is always a copula, C − is a copula only in dimension d = 2.
The comonotonic copula C + is defined as C+(u1, ···, ud) = min{u1, ···, ud}.
The lower bound C − is the function defined as
C−(u1, ···, ud) = max{0, u1 + ... + ud − (d − 1)}.
37
@freakonometrics

Fréchet-Hoeffding bounds C − and C+ , and Comonotonicity
Let X ∈ F(F1, ···, Fd). Let X+
∈ F(F1, ···, Fd) denote a random vector with
copula C+ , called comotonic version of X. In dimension d = 2, let
X −
∈ F (F 1, F 2) be a counter-comonotonic version of X .
38
@freakonometrics

Fréchet-Hoeffding bounds C − and C +
1. If d = 2, C − is the c.d.f. of (U, 1 − U ) where U ∼ 𝐶([0,1]).
2. (X1, X2 ) has copula C − if and only if there is φ strictly increasing and ψ
strictly decreasing sucht that (X1, X2 ) = (φ(Z), ψ(Z)) for some random
variable Z.
3. C + is the c.d.f. of (U, ···, U ) where U ∼ 𝐶([0,1]).
4. (X1, ···, X n ) has copula C + if and only if there are functions φi strictly
increasing such that (X1, ···, X n ) = (φ1(Z), ···, φn(Z)) for some random
variable Z.
Those bounds can be used to bound other quantities. If h : R2 → R is
2-croissante, then for any (X 1, X 2) ∈ F (F 1, F 2)
E(φ(F1
−1
(U), F2
−1
(1 − U))) ≤ E(φ(X1, X2)) ≤ E(φ(F1
−1
(U), F2
−1
(U))),
where U ∼ 𝐶([0,1]), see Tchen (1980) for more applications
39
@freakonometrics

Elliptical Copulas
Let r ∈ (−1, +1), then the Gaussian copula with parameter r (in dimension
d = 2) is
1
C (u1, u2) =
2π
√
1 − r2
∫ − 1
1
Φ (u ) ∫ 2
Φ (u )
exp
2 2
x − 2rxy + y
2(1 − r2)
dxdy
−∞ −∞
where Φ is the c.d.f. of the N (0, 1) distribution
∫ x
−∞
1 𝑥2
Φ(x) = √
2π
exp −
2
− 1
d𝑥.
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Elliptical Copulas
Let r ∈ (−1, +1), and ν > 0, then the Student t copula with parameters r and ν
is
− 1
ν 1
T (u )
∫ ∫ − 1
ν 2
T (u )
−∞ −∞
1
πν
√
1 − r2
Γ ν
2
+ 1
Γ ν
2
x − 2rxy + y
2 2
ν(1 − r2)
ν
2
− +1
1 + dxdy.
where Tν is the c.d.f. of the Student t distribution, with ν degrees of freedom
Tν(x) =
∫
−∞
2
x Γ( ν +1 )
2
√
νπ Γ( ν )
𝑥2
1 +
ν
−( ν + 1
2
)
41
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Archimedean Copulas, in dimension d = 2
Let φ denote a decreasing convex function (0, 1] → [0, ∞] such that φ(1) = 0 and
φ(0) = ∞. A (strict) Archimedean copula with generator φ is the copula defined
as
C(u1, u2) = φ−1(φ(u1) + φ(u2)), for all u1, u2 ∈ [0,1].
E.g. if φ(t) = tα − 1; this is Clayton copula.
The generator of an Archimedean copula is not unique.Further, Archimedean
copulas are symmetric, since C(u1, u2) = C(u2, u1).
42
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The only copula is radialy symmetric, i.e. C(u1, u2) = C٨(u1, u2) is such that
e−α t − 1
e−α − 1
φ(t) = log . This is Frank copula, from Frank (1979)).
Some prefer a multiplicative version of Archimedean copulas
C (u1, u2) = h−1[h(u1) ·h(u2)].
The link is h(t) = exp[φ(t)], or conversely φ(t) = h(log(t)).
43
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Remark in dimension 1, P(F (X) ≤ t) = t, i.e. F (X) ∼ 𝐶([0,1]) if X ∼ F .
Archimedean copulas can also be characterized by their Kendall function,
φ(t)
K(t) = P[C(U1, U2) ≤ t] = t − λ(t) where λ(t) =
φ'(t)
and where (U1, U2) ∼ C . Conversely,
φ(t) = exp
t
t0
ds
λ(s)
∫
,
where t0 ∈ (0, 1) is some arbitrary constant.
44
@freakonometrics

Note that Archimedean copulas can also be defined when φ(0) ≤ ∞.
Let φ denote a decreasing convex function (0, 1] → [0, ∞] such that φ(1) = 0.
Define the inverse of φ as
−1
φ (t) =
φ−1(t), for 0 ≤ t ≤ φ(0)
0, for φ(0) < t < ∞.
An Archimedean copula with generator φ is the copula defined as
C (u1, u2) = φ−1(φ(u1) + φ(u2)), for all u1, u2 ∈ [0, 1].
Non strict Archimedean copulas have a null set, {(u1, u2), φ(u1) + φ(u2) > 0} non
empty, such that
P((U1, U2) ∈ { (u1, u2), φ(u1) + φ(u2) > 0} ) = 0.
45
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This set is bounded by a null curve, {(u1, u2), φ(u1) + φ(u2) = 0}, with mass
1 2 1 2
P((U , U ) ∈ { (u , u ) 1
, φ(u ) + φ 2
(u ) = 0}) = −
φ(0)
φ'(0+)
,
which is stricly positif if −φ'(0+) < +∞.
E.g. if φ(t) = tα − 1, with α ∈ [−1, ∞), with limiting case φ(t) = − log(t) when
α = 0; this is Clayton copula.
46
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ψ ( t ) r a n g e θ
( 1 )
θ
1 − θ
( t − 1 ) [ − 1 , 0) ∪ ( 0 , ∞ ) C l a y t o n , C LAYTON ( 1 9 7 8 )
( 2 ) ( 1 − t ) θ [ 1 , ∞ )
( 3 ) l o g
1 − θ ( 1 − t )
t
[ − 1 , 1 ) A l i - M i k h a i l - H a q
( 4 ) ( − l o g t ) θ [ 1 , ∞ ) G u m b el, GUMBEL ( 1 9 6 0 ) , HOUGAARD ( 1 9 8 6 )
( 5 ) − l o g e − θ t − 1
e − θ − 1
( − ∞ , 0) ∪ ( 0 , ∞ ) F r a n k , FRANK ( 1 9 7 9 ) , NELSEN ( 1 9 8 7 )
( 6 ) − l o g { 1 − ( 1 − t ) θ } [ 1 , ∞ ) J o e, FRANK ( 1 9 8 1 ) , JOE ( 1 9 9 3 )
( 7 ) − l o g { θ t + ( 1 − θ ) } ( 0 , 1]
(8)
1 − t
1 + ( θ − 1 ) t
[ 1 , ∞ )
( 9 ) ( 0 , 1] BARNETT ( 1 9 8 0 ) , GUMBEL ( 1 9 6 0 )
( 1 0 )
l o g ( 1 − θ l o g t )
l o g ( 2 t − θ − 1 ) ( 0 , 1]
( 1 1 ) log(2 − t θ ) ( 0 , 1 / 2 ]
( 1 2 )
t
( − 1 )
1 θ [ 1 , ∞ )
( 1 3 ) ( 1 − l o g t ) θ − 1 ( 0 , ∞ )
( 1 4 ) ( t − 1 / θ − 1 ) θ [ 1 , ∞ )
( 1 5 ) ( 1 − t 1 / θ ) θ [ 1 , ∞ ) GENEST & GHOUDI ( 1 9 9 4 )
( 1 6 ) θ
t
( + 1 ) ( 1 − t ) [0, ∞ )
47
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Archimedean Copulas, in dimension d ≥ 2
Archimedean copulas are associative (see Schweizer & Sklar (1983), i.e.
C (C (u1, u2), u3) = C (u1, C (u2, u3)), for all 0 ≤ u1, u2, u3 ≤ 1.
In dimension d > 2, assume that φ−1 is d-completely monotone (where ψ is
d-completely monotine if it is continuous and for all k = 0,1, ···, d,
(−1)kdkψ(t)/dtk ≥ 0).
An Archimedean copula in dimension d ≥ 2 is defined as
C (u1, ···, un ) = φ−1(φ(u1) + ... + φ(un )), for all u1, ···, un ∈ [0, 1].
48
@freakonometrics

Those copulas are obtained iteratively, starting with
C2(u1, u2) = φ−1(φ(u1) + φ(u2))
and then, for any n ≥ 2,
C n +1(u1, ···, un +1) = C 2(C n (u1, ···, un ), un +1).
Let ψdenote the Laplace transform of a positive random variable Θ, then
(Bernstein theorem), ψis completely montone, and ψ(0) = 1. Then φ = ψ−1 is
an Archimedean generator in any dimension d ≥ 2. E.g. if Θ ∼ G(a, a), then
ψ(t) = (1 + t)1/α, and we have Clayton Clayton copula.
49
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Let X = (X1, ···, Xd ) denote remaining lifetimes, with joint survival
distribution function that is Schur-constant, i.e. there is S : R+ → [0,1] such that
P(X1 > x1, ···, X d > xd) = S(x1 + ···+ xd).
Then margins X i are also Schur-contant (i.e. exponentially distributed), and the
survival copula of X is Archimedean with generator S−1. Observe further that
P(X i − xi > t|X > x) = P(X j − xj > t|X > x),
for all t > 0and x ∈ R+
d. Hence, if S is a power function, we obtain Clayton
copula, see Nelsen (2005).
50
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Let (Cn) be a sequence of absolutely continuous Archimedean copulas, with
generators (φn). The limit of Cn , as n → ∞ is Archimedean if either
• there is a genetor φ such that s, t ∈[0, 1],
lim
n → ∞ φ'
n(t) φ'(t)
φn(s)
=
φ(s)
.
n → ∞
• there is a continuous function λ such that lim λn(t) = λ(t).
n → ∞
• there is a function K continuous such that lim Kn (t) = K(t).
n → ∞
• there is a sequence of positive constants (cn) such that lim cnφn(t) = φ(t),
for all t ∈ [0, 1].
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Copulas, Optimal Transport and Matching
Monge Kantorovich,
T :R→ R
∫
min [l (x1, T (x ))dF (
1 1 1 1
x ); wiht T (X ) 2
∼ F when X1 ∼ F1]
for some loss function l, e.g. l(x1, x2) = [x1 − x2]2.
i i
2 ٨
In the Gaussian case, if X ∼ N (0, σ ), T ( 1 2 1 1
x ) = σ /σ ·x .
Equivalently
)
∫
min l (x1, x2)dF (x1, x2) =
F ∈ F ( F 1 , F 2 F
min
∈ F ( F 1 , F 2 )
{ EF [l (X 1, X 2)]}
F ∈F (F 1 ,F 2
If l is quadratic, we want to maximize the correlation,
max { EF [X 1 ·X 2]}
)
52
@freakonometrics

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