Spherical / Elliptical Distributions, Distributions on the Simplex, and Copulas

1. Arthur 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
@freakonometrics 1

2. Arthur CHARPENTIER - Multivariate Distributions
Geometry in Rd
and Statistics
The standard inner product is < x, y > 2 = xT
y = i xiyi.
Hence, x ⊥ y if < x, y > 2
= 0.
The Euclidean norm is x 2
=< x, x >
1
2
2
=
n
i=1 x2
i
1
2
.
The unit sphere of Rd
is Sd = {x ∈ Rd
: x 2 = 1}.
If x = {x1, · · · , xn}, note that the empirical covariance is
Cov(x, y) =< x − x, y − y > 2
and Var(x) = x − x 2
.
For the (multivariate) linear model, yi = β0 + βT
1 xi + εi, or equivalently,
yi = β0+ < β1, xi > 2
+εi
@freakonometrics 2

3. Arthur CHARPENTIER - Multivariate Distributions
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
some lower triangular matrix L satisfying Σ = LLT
. Denote L = Σ
1
2 .
With Cholesky decomposition, we have the particular case (with a Gaussian
distribution) of Rosenblatt (1952)’s chain,
f(x1, x2, · · · , xd) = f1(x1) · f2|1(x2|x1) · f3|2,1(x3|x2, x1) · · ·
· · · fd|d−1,··· ,2,1(xd|xd−1, · · · , x2, x1).
f(x; µ, Σ) =
1
(2π)
d
2 |Σ|
1
2
exp −
1
2
(x − µ)T
Σ−1
(x − µ)
x µ,Σ
for all x ∈ Rd
.
@freakonometrics 3

4. Arthur CHARPENTIER - Multivariate Distributions
The d dimensional Gaussian Random Vector
Note that x µ,Σ = (x − µ)T
Σ−1
(x − µ) is the Mahalanobis distance.
Define the ellipsoid Eµ,Σ = {x ∈ Rd
: x µ,Σ = 1}
Let
X =


X1
X2

 ∼ N




µ1
µ2

 ,


Σ11 Σ12
Σ21 Σ22




then
X1|X2 = x2 ∼ N(µ1 + Σ12Σ−1
22 (x2 − µ2) , Σ11 − Σ12Σ−1
22 Σ21)
X1 ⊥⊥ X2 if and only if Σ12 = 0.
Further, if X ∼ N(µ, Σ), then AX + b ∼ N(Aµ + b, AΣAT
).
@freakonometrics 4

5. Arthur CHARPENTIER - Multivariate Distributions
The Gaussian Distribution, as a Spherical Distributions
If X ∼ N(0, I), then X = R · U, where
R2
= X 2
∼ χ2
(d)
and
U = X/ X 2
∼ U(Sd),
with R ⊥⊥ U.
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
q
@freakonometrics 5

6. Arthur CHARPENTIER - Multivariate Distributions
The Gaussian Distribution, as an Elliptical Distributions
If X ∼ N(µ, Σ), then X = µ + R · Σ
1
2 · U, where
R2
= X 2
∼ χ2
(d)
and
U = X/ X 2 ∼ U(Sd),
with R ⊥⊥ U.
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
q
@freakonometrics 6

7. Arthur CHARPENTIER - Multivariate Distributions
Spherical Distributions
Let M denote an orthogonal matrix, MT
M = MMT
= I. X has a spherical
distribution if X
L
= MX.
E.g. in R2
, 

cos(θ) − sin(θ)
sin(θ) cos(θ)




X1
X2

 L
=


X1
X2


For every a ∈ Rd
, aT
X
L
= a 2
· Xi, for any i ∈ {1, · · · , d}.
Further, the generating function of X can be written
E[eitT
X
] = ϕ(tT
t) = ϕ( t 2
2
), ∀t ∈ Rd
,
for some ϕ : R+ → R+.
@freakonometrics 7

8. Arthur CHARPENTIER - Multivariate Distributions
Uniform Distribution on the Sphere
Actually, more complex that it seems...



x1 = ρ sin ϕ cos θ
x2 = ρ sin ϕ sin θ
x3 = ρ cos ϕ
with ρ > 0, ϕ ∈ [0, 2π] and θ ∈ [0, π].
If Φ ∼ U([0, 2π]) and Θ ∼ U([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
@freakonometrics 8

9. Arthur CHARPENTIER - Multivariate Distributions
Spherical Distributions
Random vector X as a spherical distribution if
X = R · U
where R is a positive random variable and U is uniformly
distributed on the unit sphere of Rd
, Sd, with R ⊥⊥ U
E.g. X ∼ N(0, I).
−2 −1 0 1 2
−2−1012
q
q
q q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
qq
q
q
qq
q
q
qq
q
q
q
q
q
q
q
q
q
q
qq
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
qq
q q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q q q q
q
q
q
qq
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
−2 −1 0 1 2
−2−1012
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
0.02
0.04
0.06
0.08
0.12
0.14
@freakonometrics 9

10. Arthur CHARPENTIER - Multivariate Distributions
Elliptical Distributions
Random vector X as a elliptical distribution if
X = µ + R · A · U
where A satisfies AA = Σ, U(Sd), with R ⊥⊥ U. Denote
Σ
1
2 = A.
E.g. X ∼ N(µ, Σ).
−2 −1 0 1 2
−2−1012
q
q
q
q
qq
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
qq
q
q
q
q
q
q
qq
q
q
qq
q
q
q
q
q
q
qq
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
qq
qq
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q q
qq
q q
q
q
q
q
q q
q
q
q
q
q q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
qq
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
−2 −1 0 1 2
−2−1012
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
qq
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
0.02
0.04
0.06
0.08
0.12
0.14
@freakonometrics 10

11. Arthur CHARPENTIER - Multivariate Distributions
Elliptical Distributions
X = µ + RΣ
1
2 U where R is a positive random variable, U ∼ U(Sd), with
U ⊥⊥ R. If X ∼ FR, then X ∼ E(µ, Σ, FR).
Remark Instead of FR it is more common to use ϕ such that
E[eitT
X
] = eitT
µ
ϕ(tT
Σt), t ∈ Rd
.
E[X] = µ and Var[X] = −2ϕ (0)Σ
f(x) ∝
1
|Σ|
1
2
f( (x − µ)TΣ−1
(x − µ))
where f : R+ → R+ is called radial density. Note that
dF(r) ∝ rd−1
f(r)1(x > 0).
@freakonometrics 11

12. Arthur CHARPENTIER - Multivariate Distributions
Elliptical Distributions
If X ∼ E(µ, Σ, FR), then
AX + b ∼ E(Aµ + b, AΣAT
, FR)
If
X =


X1
X2

 ∼ E




µ1
µ2

 ,


Σ11 Σ12
Σ21 Σ22

 , FR


then
X1|X2 = x2 ∼ E(µ1 + Σ12Σ−1
22 (x2 − µ2) Σ11 − Σ12Σ−1
22 Σ21, F1|2)
where
F1|2 is the c.d.f. of (R2
− )
1
2 given X2 = x2.
@freakonometrics 12

13. Arthur CHARPENTIER - Multivariate Distributions
Mixtures of Normal Distributions
Let Z ∼ N(0, I). Let W denote a positive random variable, Z ⊥⊥ W. Set
X = µ +
√
WΣ
1
2 Z,
so that X|W = w ∼ N(µ, wΣ).
E[X] = µ and Var[X] = E[W]Σ
E[eitT
X
] = E eitT
µ− 1
2 W tT
Σt)
, t ∈ Rd
.
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.
@freakonometrics 13

14. Arthur CHARPENTIER - Multivariate Distributions
Multivariate Student t
X ∼ t(µ, Σ, ν) if
X = µ + Σ
1
2
Z
√
W/ν
where Z ∼ N(0, I) and W ∼ χ2
(ν), with Z ⊥⊥ W.
Note that
Var[X] =
ν
ν − 2
Σ if ν > 2.
@freakonometrics 14

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

16. Arthur CHARPENTIER - Multivariate Distributions
On Conditional Independence, de Finetti & Hewitt
Instead of X
L
= MX 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(Xi, Xj) ≥ −
1
d − 1
.
From de Finetti (1931), X1, · · · , Xd, · · · are exchangeable {0, 1} variables if and
only if there is a c.d.f. Π on [0, 1] such that
P[X = x] =
1
0
θxT
1
[1 − θ]n−xT
1
dΠ(θ),
i.e. X1, · · · , Xd, · · · are (conditionnaly) independent given Θ ∼ Π.
@freakonometrics 16

17. Arthur CHARPENTIER - Multivariate Distributions
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
P[Ti > t|Θ = θ] = F
θ
i,0(t)
Assume that lifetimes are (conditionnaly) independent given Θ.
@freakonometrics 17

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

19. Arthur CHARPENTIER - Multivariate Distributions
Compositional Data and Geometry of the Simplex
Following Aitchison (1986), given x ∈ Rd
+ define the closure operator C
C[x1, x2, · · · , xd] =
x1
d
i=1 xi
,
x2
d
i=1 xi
, . . . ,
xd
d
i=1 xi
∈ Sd.
It is possible to define (Aitchison) inner product on Sd
< x, y >a=
1
2d i,j
log
xi
xj
log
yi
yj
=
i
log
xi
x
log
yi
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.
@freakonometrics 19

20. Arthur CHARPENTIER - Multivariate Distributions
Dirichlet Distributions
Given α ∈ Rd
+, and x ∈ Sd ⊂ Rd
f (x1, · · · , xd; α) =
1
B(α)
d
i=1
xαi−1
i ,
where
B(α) =
d
i=1 Γ(αi)
Γ
d
i=1 αi
Then
Xi ∼ Beta αi,
d
j=1
αj − αi .
and E(X) = C(α).
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqq
qqqqqqqqqq
qqqqqqqqq
qqqqqqqqq
qqqqqqq
qqqqqqq
qqqqq
qqqq
qqqq
qqqqq
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
@freakonometrics 20

21. Arthur CHARPENTIER - Multivariate Distributions
Dirichlet Distributions
Stochastic Representation
Let Z = (Z1, · · · , Zd) denote independent G(αi, θ) random
variables. Then S = Z1 + · · · + Zd = ZT
1 has a G(αT
1, θ)
distribution, and
X = C(X) =
Z
S
=
Z1
d
i=1 Zi
, · · · ,
Zd
d
i=1 Zi
has a Dirichlet distribution Dirichlet(α).
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqq
qqqqqqqqqq
qqqqqqqqq
qqqqqqqqq
qqqqqqq
qqqqqqq
qqqqq
qqqq
qqqq
qqqqq
q
q
q
q
q
q
q
q
q
q
q q qq
q
q
q
q
q
q
@freakonometrics 21

22. Arthur CHARPENTIER - Multivariate Distributions
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 U([0, 1])
distribution. Define spacings, as
Xi = U(i−1):d − U where Ui:d are order statistics
with conventions U0:d = 0 and Ud:d = 1. Then
X = (X1, · · · , Xd) ∼ U(Sd).
@freakonometrics 22

23. Arthur CHARPENTIER - Multivariate Distributions
‘Normal distribution on the Simplex’
(also called logistic-normal).
Let ˜Y ∼ N(µ, Σ) in dimension d − 1. Set Z = ( ˜Y , 0) and
X = C(eZ
) =
eZ1
eZ1 + · · · + eZd
, · · · ,
eZd
eZ1 + · · · + eZd
@freakonometrics 23

24. Arthur CHARPENTIER - Multivariate Distributions
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)
P[X ∈ A] =
d
i=1
P[Xi ∈ Ai], where A = A1 × · · · , ×Ad.
E.g. if Ai = (−∞, xi), then
F(x) = P[X ∈ (−∞, x] =
d
i=1
P[Xi ∈ (−∞, xi] =
d
i=1
Fi(xi).
If F is absolutely continous,
f(x) =
∂d
F(x)
∂x1 · · · ∂xd
=
d
i=1
fi(xi).
@freakonometrics 24

25. Arthur CHARPENTIER - Multivariate Distributions
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 Xi ∼ Fi.
Note that for any F ∈ F(F1, · · · , Fd), ∀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).
@freakonometrics 25

26. Arthur CHARPENTIER - Multivariate Distributions
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) = 0 for 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.
@freakonometrics 26

27. Arthur CHARPENTIER - Multivariate Distributions
Copulas in Dimension 2
Borders of the copula function
!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
!0.20.00.20.40.60.81.01.21.4
!0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Border conditions, in dimension d = 2, C(u1, 0) = C(0, u2) = 0, C(u1, 1) = u1 et
C(1, u2) = u2.
@freakonometrics 27

28. Arthur CHARPENTIER - Multivariate Distributions
Copulas in Dimension 2
If C is the copula of random vector (X1, X2), then C couples marginal
distributions, in the sense that
P(X1 ≤ x1, X2 ≤ x2) = C(P(X1 ≤ x1),P(X2 ≤ x2))
Note tht is is also possible to couple survival distributions: there exists a copula
C such that
P(X > x, Y > y) = C (P(X > x), P(Y > y)).
Observe that
C (u1, u2) = u1 + u2 − 1 + C(1 − u1, 1 − u2).
The survival copula C associated to C is the copula defined by
C (u1, u2) = u1 + u2 − 1 + C(1 − u1, 1 − u2).
Note that (1 − U1, 1 − U2) ∼ C if (U1, U2) ∼ C.
@freakonometrics 28

29. Arthur CHARPENTIER - Multivariate Distributions
Copulas in Dimension 2
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−1
2 (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(F−1
1 (u1), F−1
2 (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|.
@freakonometrics 29

30. Arthur CHARPENTIER - Multivariate Distributions
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 ≤ Xd ≤ 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
Vh ([a, b]) = ∆b
ah (t) = ∆bd
ad
∆bd−1
ad−1
...∆b2
a2
∆b1
a1
h (t)
for any t, where
∆bi
ai
h (t) = h (t1, · · · , ti−1, bi, ti+1, · · · , tn) − h (t1, · · · , ti−1, ai, ti+1, · · · , tn) .
@freakonometrics 30

31. Arthur CHARPENTIER - Multivariate Distributions
Copulas in Dimension d
Black dot, + sign, white dot, - sign.
@freakonometrics 31

32. Arthur CHARPENTIER - Multivariate Distributions
Copulas in Dimension d
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).
@freakonometrics 32

33. Arthur CHARPENTIER - Multivariate Distributions
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(F1(x1), · · · , Fd(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
,
C(u1, · · · , ud) = F(F−1
1 (u1), · · · , F−1
d (ud)),
where F−1
1 , · · · , F−1
n are generalized quantiles.
@freakonometrics 33

34. Arthur CHARPENTIER - Multivariate Distributions
Copulas in Dimension d
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(X1), · · · , φd(Xd)).
If C is a copula, then function
C (u1, · · · , ud) =
d
k=0

(−1)k
i1,··· ,ik
C(1, · · · , 1, 1 − ui1 , 1, ...1, 1 − uik
, 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 . And if
P(X1 ≤ x1, · · · , Xd ≤ xd) = C(P(X1 ≤ x1), · · · , P(Xd ≤ xd)),
for all (x1, · · · , xd) ∈ R, then
P(X1 > x1, · · · , Xd > xd) = C (P(X1 > x1), · · · , P(Xd > xd)).
@freakonometrics 34

35. Arthur CHARPENTIER - Multivariate Distributions
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 C be 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).
@freakonometrics 35

36. Arthur CHARPENTIER - Multivariate Distributions
The Indepedent Copula C⊥
, or Π
The independent copula C⊥
is the copula defined as
C⊥
(u1, · · · , un) = u1 · · · ud =
d
i=1
ui (= Π(u1, · · · , un)).
Let X ∈ F(F1, · · · , Fd), then X⊥
∈ F(F1, · · · , Fd) will denote a random vector
with copula C⊥
, called ‘independent version’ of X.
@freakonometrics 36

37. Arthur CHARPENTIER - Multivariate Distributions
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)}.
@freakonometrics 37

38. Arthur CHARPENTIER - Multivariate Distributions
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(F1, F2) be a counter-comonotonic version of X.
@freakonometrics 38

39. Arthur CHARPENTIER - Multivariate Distributions
Fréchet-Hoeffding bounds C−
and C+
1. If d = 2, C−
is the c.d.f. of (U, 1 − U) where U ∼ U([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 ∼ U([0, 1]).
4. (X1, · · · , Xn) has copula C+
if and only if there are functions φi strictly
increasing such that (X1, · · · , Xn) = (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 (X1, X2) ∈ F(F1, F2)
E(φ(F−1
1 (U), F−1
2 (1 − U))) ≤ E(φ(X1, X2)) ≤ E(φ(F−1
1 (U), F−1
2 (U))),
where U ∼ U([0, 1]), see Tchen (1980) for more applications
@freakonometrics 39

40. Arthur CHARPENTIER - Multivariate Distributions
Elliptical Copulas
Let r ∈ (−1, +1), then the Gaussian copula with parameter r (in dimension
d = 2) is
C(u1, u2) =
1
2π
√
1 − r2
Φ−1
(u1)
−∞
Φ−1
(u2)
−∞
exp
x2
− 2rxy + y2
2(1 − r2)
dxdy
where Φ is the c.d.f. of the N(0, 1) distribution
Φ(x) =
x
−∞
1
√
2π
exp −
z2
2
dz.
@freakonometrics 40

41. Arthur CHARPENTIER - Multivariate Distributions
Elliptical Copulas
Let r ∈ (−1, +1), and ν > 0, then the Student t copula with parameters r and ν
is
T −1
ν (u1)
−∞
T −1
ν (u2)
−∞
1
πν
√
1 − r2
Γ ν
2 + 1
Γ ν
2
1 +
x2
− 2rxy + y2
ν(1 − r2)
− ν
2 +1
dxdy.
where Tν is the c.d.f. of the Student t distribution, with ν degrees of freedom
Tν(x) =
x
−∞
Γ(ν+1
2 )
√
νπ Γ(ν
2 )
1 +
z2
ν
−( ν+1
2 )
@freakonometrics 41

42. Arthur CHARPENTIER - Multivariate Distributions
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).
@freakonometrics 42

43. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d = 2
The only copula is radialy symmetric, i.e. C(u1, u2) = C (u1, u2) is such that
φ(t) = log
e−αt
− 1
e−α − 1
. 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)).
@freakonometrics 43

44. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d = 2
Remark in dimension 1, P(F(X) ≤ t) = t, i.e. F(X) ∼ U([0, 1]) if X ∼ F.
Archimedean copulas can also be characterized by their Kendall function,
K(t) = P[C(U1, U2) ≤ t] = t − λ(t) where λ(t) =
φ(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.
@freakonometrics 44

45. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d = 2
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
φ−1
(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.
@freakonometrics 45

46. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d = 2
This set is bounded by a null curve, {(u1, u2), φ(u1) + φ(u2) = 0}, with mass
P((U1, U2) ∈ {(u1, u2), φ(u1) + φ(u2) = 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.
@freakonometrics 46

47. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d = 2
ψ(t) range θ
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)
(2) (1 − t)θ [1, ∞)
(3) log
1−θ(1−t)
t
[−1, 1) Ali-Mikhail-Haq
(4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986)
(5) − log
e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987)
(6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993)
(7) − log{θt + (1 − θ)} (0, 1]
(8)
1−t
1+(θ−1)t
[1, ∞)
(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)
(10) log(2t−θ − 1) (0, 1]
(11) log(2 − tθ) (0, 1/2]
(12) ( 1
t
− 1)θ [1, ∞)
(13) (1 − log t)θ − 1 (0, ∞)
(14) (t−1/θ − 1)θ [1, ∞)
(15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994)
(16) ( θ
t
+ 1)(1 − t) [0, ∞)
@freakonometrics 47

48. Arthur CHARPENTIER - Multivariate Distributions
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)k
dk
ψ(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].
@freakonometrics 48

49. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d ≥ 2
Those copulas are obtained iteratively, starting with
C2(u1, u2) = φ−1
(φ(u1) + φ(u2))
and then, for any n ≥ 2,
Cn+1(u1, · · · , un+1) = C2(Cn(u1, · · · , un), un+1).
Let ψ denote the Laplace transform of a positive random variable Θ, then
(Bernstein theorem), ψ is 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.
@freakonometrics 49

50. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d ≥ 2
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, · · · , Xd > xd) = S(x1 + · · · + xd).
Then margins Xi are also Schur-contant (i.e. exponentially distributed), and the
survival copula of X is Archimedean with generator S−1
. Observe further that
P(Xi − xi > t|X > x) = P(Xj − xj > t|X > x),
for all t > 0 and x ∈ Rd
+. Hence, if S is a power function, we obtain Clayton
copula, see Nelsen (2005).
@freakonometrics 50

51. Arthur CHARPENTIER - Multivariate Distributions
Archimedean Copulas, in dimension d ≥ 2
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→∞
φn(s)
φn(t)
=
φ(s)
φ (t)
.
• there is a continuous function λ such that lim
n→∞
λn(t) = λ(t).
• there is a function K continuous such that lim
n→∞
Kn(t) = K(t).
• there is a sequence of positive constants (cn) such that lim
n→∞
cnφn(t) = φ(t),
for all t ∈ [0, 1].
@freakonometrics 51

52. Arthur CHARPENTIER - Multivariate Distributions
Copulas, Optimal Transport and Matching
Monge Kantorovich,
min
T :R→R
[ (x1, T(x1))dF1(x1); wiht T(X1) ∼ F2 when X1 ∼ F1]
for some loss function , e.g. (x1, x2) = [x1 − x2]2
.
In the Gaussian case, if Xi ∼ N(0, σ2
i ), T (x1) = σ2/σ1 · x1.
Equivalently
min
F ∈F(F1,F2)
(x1, x2)dF(x1, x2) = min
F ∈F(F1,F2)
{EF [ (X1, X2)]}
If is quadratic, we want to maximize the correlation,
max
F ∈F(F1,F2)
{EF [X1 · X2]}
@freakonometrics 52