Multivariate Distributions, an overview

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


Geometry in Rd and Statistics

The standard inner product is < x,y >`2= xTy =∑i xiyi.

Hence, x ⊥ y if < x,y >`2= 0.

The Euclidean norm is ‖x‖`2 =< x,x >12`2

=(∑n

i=1 x2i

) 12 .

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 + βT1xi + εi, or equivalently,

yi = β0+ < β1,xi >`2 +εi

@freakonometrics 2


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 forsome lower triangular matrix L satisfying Σ = LLT. Denote L = Σ

12 .

With Cholesky decomposition, we have the particular case (with a Gaussiandistribution) 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) d2 |Σ| 12

exp(− 1

2 (x− µ)TΣ−1(x− µ)︸︷︷︸‖x‖µ,Σ

)for all x ∈ Rd.

@freakonometrics 3


The d dimensional Gaussian Random Vector

Note that ‖x‖µ,Σ = (x− µ)TΣ−1(x− µ) is the Mahalanobis distance.

Define the ellipsoid Eµ,Σ = {x ∈ Rd : ‖x‖µ,Σ = 1}

Let

X =

X1

X2

∼ Nµ1

µ2

,Σ11 Σ12

Σ21 Σ22

then

X1|X2 = x2 ∼ N (µ1 + Σ12Σ−122 (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


The Gaussian Distribution, as a Spherical Distributions

If X ∼ N (0, I), then X = R ·U , where

R2 = ‖X‖`2 ∼ χ2(d)

andU = X/‖X‖`2 ∼ U(Sd),

with R ⊥⊥ U .

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@freakonometrics 5


The Gaussian Distribution, as an Elliptical Distributions

If X ∼ N (µ,Σ), then X = µ+R ·Σ12 ·U︸︷︷︸, where

R2 = ‖X‖`2 ∼ χ2(d)

andU = X/‖X‖`2 ∼ U(Sd),

with R ⊥⊥ U .

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@freakonometrics 6


Spherical Distributions

Let M denote an orthogonal matrix, MTM = MMT = I. X has a sphericaldistribution 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, aTX

L= ‖a‖`2 ·Xi, for any i ∈ {1, · · · , d}.

Further, the generating function of X can be written

E[eitTX ] = ϕ(tTt) = ϕ(‖t‖2`2

), ∀t ∈ Rd,

for some ϕ : R+ → R+.

@freakonometrics 7


Uniform Distribution on the Sphere

Actually, more complex that it seems...x1 = ρ sinϕ cos θx2 = ρ sinϕ sin θx3 = ρ cosϕ

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


Spherical Distributions

Random vector X as a spherical distribution if

X = R ·U

where R is a positive random variable and U is uniformlydistributed on the unit sphere of Rd, Sd, with R ⊥⊥ U

E.g. X ∼ N (0, I).

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@freakonometrics 9


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 . DenoteΣ

12 = A.

E.g. X ∼ N (µ,Σ).

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@freakonometrics 10



X = µ+RΣ12U 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[eitTX ] = eit

Tµϕ(tTΣt), t ∈ Rd.

E[X] = µ and Var[X] = −2ϕ′(0)Σ

f(x) ∝ 1|Σ| 12

f(√

(x− µ)TΣ−1(x− µ))

where f : R+ → R+ is called radial density. Note that

dF (r) ∝ rd−1f(r)1(x > 0).

@freakonometrics 11



If X ∼ E(µ,Σ, FR), then

AX + b ∼ E(Aµ+ b,AΣAT, FR)

If

X =

X1

X2

∼ Eµ1

µ2

,Σ11 Σ12

Σ21 Σ22

, FR

then

X1|X2 = x2 ∼ E(µ1 + Σ12Σ−122 (x2 − µ2) Σ11 −Σ12Σ−1

22 Σ21, F1|2)

whereF1|2 is the c.d.f. of (R2 − ?) 1

2 given X2 = x2.

@freakonometrics 12


Mixtures of Normal Distributions

Let Z ∼ N (0, I). Let W denote a positive random variable, Z ⊥⊥W . Set

X = µ+√WΣ

12Z,

so that X|W = w ∼ N (µ, wΣ).

E[X] = µ and Var[X] = E[W ]Σ

E[eitTX ] = E

[eit

Tµ− 12W t

TΣt)], 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 amultivariate t distribution, with ν degrees of freedom.

@freakonometrics 13


Multivariate Student t

X ∼ t(µ,Σ, ν) if

X = µ+ Σ12

Z√W/ν

where Z ∼ N (0, I) and W ∼ χ2(ν), with Z ⊥⊥W .

Note thatVar[X] = ν

ν − 2Σ if ν > 2.

@freakonometrics 14


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


On Conditional Independence, de Finetti & Hewitt

Instead of X L= MX for any orthogonal matrix M , consider the equality for anypermutation 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 6= j. Note that necessarily

ρ = Corr(Xi, Xj) ≥ −1

d− 1 .

From de Finetti (1931), X1, · · · , Xd, · · · are exchangeable {0, 1} variables if andonly if there is a c.d.f. Π on [0, 1] such that

P[X = x] =∫ 1

0θx

T1[1− θ]n−xT1dΠ(θ),

i.e. X1, · · · , Xd, · · · are (conditionnaly) independent given Θ ∼ Π.

@freakonometrics 16


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-typeproportional hazard µi(t) = Θ · µi,0(t), so that

P[Ti > t|Θ = θ] = Fθ

i,0(t)

Assume that lifetimes are (conditionnaly) independent given Θ.

@freakonometrics 17


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 thatSd = {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− xT1) ∈ 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


Compositional Data and Geometry of the Simplex

Following Aitchison (1986), given x ∈ Rd+ define the closure operator C

C[x1, x2, · · · , xd] =[

x1∑di=1 xi

,x2∑di=1 xi

, . . . ,xd∑di=1 xi

]∈ Sd.

It is possible to define (Aitchison) inner product on Sd

< x,y >a= 12d∑i,j

log xixj

log yiyj

=∑i

log xix

log yiy

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


Dirichlet Distributions

Given α ∈ Rd+, and x ∈ Sd ⊂ Rd

f (x1, · · · , xd;α) = 1B(α)

d∏i=1

xαi−1i ,

where

B(α) =∏di=1 Γ(αi)

Γ(∑d

i=1 αi

)Then

Xi ∼ Beta(αi,

(∑d

j=1αj

)− αi

).

and E(X) = C(α).

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@freakonometrics 20


Dirichlet Distributions

Stochastic RepresentationLet Z = (Z1, · · · , Zd) denote independent G(αi, θ) randomvariables. Then S = Z1 + · · ·+ Zd = ZT1 has a G(αT1, θ)distribution, and

X = C(X) = Z

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@freakonometrics 21


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


‘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


Distribution on Rd or [0, 1]d

Technically, things are more simple when X = (X1, · · · , Xd) take values in aproduct 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) = ∂dF (x)∂x1 · · · ∂xd

=d∏i=1

fi(xi).

@freakonometrics 24


Fréchet classes

Given some (univariate) cumulative distribution functions F1, · · · , Fd R→ [0, 1],let F(F1, · · · , Fd) denote the set of multivariate cumulative distribution functionof random vectors X such that Xi ∼ Fi.

Note that for any F ∈ F(F1, · · · , Fd), ∀x ∈ Rd,

F−(x) ≤ F (x) ≤ F+(x)

whereF+(x) = min{Fi(xi), i = 1, · · · , d},

andF−(x) = max{0, F1(x1) + · · ·+ Fd(xd)− (d− 1)}.

Note that F+ ∈ F(F1, · · · , Fd), while usually F− /∈ F(F1, · · · , Fd).

@freakonometrics 25


Copulas in Dimension 2

A copula C : [0, 1]2 → [0, 1] is a cumulative distribution function with uniformmargins 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



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 1.2

Border conditions, in dimension d = 2, C(u1, 0) = C(0, u2) = 0, C(u1, 1) = u1 etC(1, u2) = u2.

@freakonometrics 27



If C is the copula of random vector (X1, X2), then C couples marginaldistributions, 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 copulaC? such that

P(X > x, Y > y) = C?(P(X > x),P(Y > y)).

Observe thatC?(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



If X has distribution F ∈ F(F1, F2), with absolutely continuous margins, thenits copula is

C(u1, u2) = F (F−11 (u1), F−1

2 (u2)), ∀u1, u2 ∈ [0, 1].

More generally, if h−1 denotes the generalized inverse of some increasing functionh : R→ R, defined as h−1(t) = inf{x, h(x) ≥ t, t ∈ R}, thenC(u1, u2) = F (F−1

1 (u1), F−12 (u2)) is one copula of X.

Note that copulas are continuous functions; actually they are Lipschitz: for all0 ≤ ui, vi ≤ 1,

|C(u1, u2)− C(v1, v2)| ≤ |u1 − v1|+ |u2 − v2|.

@freakonometrics 29


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]) = ∆bah (t) = ∆bd

ad∆bd−1ad−1

...∆b2a2

∆b1a1h (t)

for any t, where

∆biaih (t) = h (t1, · · · , ti−1, bi, ti+1, · · · , tn)− h (t1, · · · , ti−1, ai, ti+1, · · · , tn) .

@freakonometrics 30



Black dot, + sign, white dot, - sign.

@freakonometrics 31



A copula in dimension d is a cumulative distribution function on [0, 1]d withuniform 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


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 theequation above holds. This function is not unique, but it is if marginsF1, · · · , Fd are absolutely continousand then, for any (u1, · · · , ud) ∈ [0, 1]d,

C(u1, · · · , ud) = F (F−11 (u1), · · · , F−1

d (ud)),

where F−11 , · · · , F−1

n are generalized quantiles.

@freakonometrics 33



Let (X1, · · · , Xd) be a random vector with copula C. Let φ1, · · · , φd, φi : R→ Rdenote 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


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 7→ 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 upperbounds 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 ofChoquet capacities as lower bounds of probability measures).

@freakonometrics 35


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 vectorwith copula C⊥, called ‘independent version’ of X.

@freakonometrics 36


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)}

andC+(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 asC−(u1, · · · , ud) = max{0, u1 + ...+ ud − (d− 1)}.

@freakonometrics 37


Fréchet-Hoeffding bounds C− and C+, and Comonotonicity

Let X ∈ F(F1, · · · , Fd). Let X+ ∈ F(F1, · · · , Fd) denote a random vector withcopula C+, called comotonic version of X. In dimension d = 2, letX− ∈ F(F1, F2) be a counter-comonotonic version of X.

@freakonometrics 38


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 ψstrictly decreasing sucht that (X1, X2) = (φ(Z), ψ(Z)) for some randomvariable 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 strictlyincreasing such that (X1, · · · , Xn) = (φ1(Z), · · · , φn(Z)) for some randomvariable Z.

Those bounds can be used to bound other quantities. If h : R2 → R is2-croissante, then for any (X1, X2) ∈ F(F1, F2)

E(φ(F−11 (U), F−1

2 (1− U))) ≤ E(φ(X1, X2)) ≤ E(φ(F−11 (U), F−1

2 (U))),

where U ∼ U([0, 1]), see Tchen (1980) for more applications

@freakonometrics 39


Elliptical Copulas

Let r ∈ (−1,+1), then the Gaussian copula with parameter r (in dimensiond = 2) is

C(u1, u2) = 12π√

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(−z

2

2

)dz.

@freakonometrics 40


Elliptical Copulas

Let r ∈ (−1,+1), and ν > 0, then the Student t copula with parameters r and νis ∫ T−1

ν (u1)

−∞

∫ T−1ν (u2)

−∞

1πν√

1− r2

Γ(ν2 + 1

)Γ(ν2) (

1 + x2 − 2rxy + y2

ν(1− r2)

)− ν2 +1

dxdy.

where Tν is the c.d.f. of the Student t distribution, with ν degrees of freedom

Tν(x) =∫ x

−∞

Γ(ν+12 )

√νπ Γ(ν2 )

(1 + z2

ν

)−( ν+12 )

@freakonometrics 41


Archimedean Copulas, in dimension d = 2

Let φ denote a decreasing convex function (0, 1]→ [0,∞] such that φ(1) = 0 andφ(0) =∞. A (strict) Archimedean copula with generator φ is the copula definedas

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, Archimedeancopulas are symmetric, since C(u1, u2) = C(u2, u1).

@freakonometrics 42



The only copula is radialy symmetric, i.e. C(u1, u2) = C?(u1, u2) is such that

φ(t) = log e−αt − 1e−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



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



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} nonempty, such that

P((U1, U2) ∈ {(u1, u2), φ(u1) + φ(u2) > 0}) = 0.

@freakonometrics 45



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) whenα = 0; this is Clayton copula.

@freakonometrics 46


Archimedean Copulas, in dimension d = 2ψ(t) range θ

(1) 1θ

(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−1e−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−t1+(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) ( 1t− 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) ( θt

+ 1)(1 − t) [0,∞)

@freakonometrics 47


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 isd-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].

@freakonometrics 48



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 isan Archimedean generator in any dimension d ≥ 2. E.g. if Θ ∼ G(a, a), thenψ(t) = (1 + t)1/α, and we have Clayton Clayton copula.

@freakonometrics 49



Let X = (X1, · · · , Xd) denote remaining lifetimes, with joint survivaldistribution 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 thesurvival 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 Claytoncopula, see Nelsen (2005).

@freakonometrics 50



Let (Cn) be a sequence of absolutely continuous Archimedean copulas, withgenerators (φn). The limit of Cn, as n→∞ is Archimedean if either

• there is a genetor φ such that s, t ∈ [0, 1],

limn→∞

φn(s)φ′n(t) = φ(s)

φ′(t) .

• there is a continuous function λ such that limn→∞

λn(t) = λ(t).

• there is a function K continuous such that limn→∞

Kn(t) = K(t).

• there is a sequence of positive constants (cn) such that limn→∞

cnφn(t) = φ(t),for all t ∈ [0, 1].

@freakonometrics 51


Copulas, Optimal Transport and Matching

Monge Kantorovich,

minT :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, σ2i ), T ?(x1) = σ2/σ1 · x1.

Equivalently

minF∈F(F1,F2)

{∫`(x1, x2)dF (x1, x2)

}= minF∈F(F1,F2)

{EF [`(X1, X2)]}

If ` is quadratic, we want to maximize the correlation,

maxF∈F(F1,F2)

{EF [X1 ·X2]}

@freakonometrics 52

Economy & Finance

Multivariate Distributions, an overview