Elliptical distributions

The class of the elliptical distributions provides a rich source of multivariate distributions that share many of the tractable properties of the multivariate Gaussian distribution. We begin with a definition and characterization of the sub-class of the spherical distributions. Then we proceed with the elliptical distributions.

Definition 4. Let X be a n-dimensional random vector. X is said to be spherically distributed, or simply spherical, if and only if X =d OX for every n×northonormal matrix O and =d denotes equality in distribution.

The class of the spherical distributions is rotationally symmetric accord- ing to the definition. This is because orthonormal transforms of vectors do not change their norms but just their orientation. Let U be uniformly dis- tributed on the unit sphereSn_,Sn_:={x∈Rn_{:||x||= 1}where||x||denotes}

the Euclidean norm. Then every random vectorXwhich can be represented asX =d RU where R is a non-negative random variable independent of U, is rotationally symmetric and thus spherical. The statement also appears correct if considered in reverse direction, any spherical random vector is nec- essarily representable byRU and R is called thegenerating random variate

of X.

It is possible to characterize the spherical distributions in terms of their characteristic function ϕX(t) = Eexp(iht, Xi) where t ∈ Rn and ht, Xi =

Pn

k=1tkXk denotes the dot product.

Theorem 2. A random vector X is spherically distributed if and only if its characteristic function has the form ϕX(t) =φX(tTt).

The function φX(·) is called characteristic generator of X. This repre-

sentation implies a simple form of the characteristic function of any affine transformation of a spherical distribution.

Proposition 1. Let X be a k-dimensional spherically distributed ran- dom vector with characteristic generator φX(·). Further, let Λ ∈ Rn×k

be an arbitrary matrix and µ _∈ Rn_{. Then the characteristic function of}

Y =µ+ ΛX has the form

ϕY(t) = exp(itTµ)φX(tTΣt), t∈Rn

where Σ = ΛΛT.

Proof. The statement is verified directly using the special representation of the characteristic function of spherical random vectors.

Definition 5. If X is a n-dimensional random vector and, for some

µ _∈ Rn _{and some n×n nonnegative definite, symmetric matrix Σ, the}

characteristic functionϕX−µ(t) ofX−µis a function of the quadratic form

tTΣt, ϕX−µ(t) =φ(tTΣt), we say that X has an elliptical distribution with

parameters µ, Σ and φ, and we write X_∈En(µ,Σ, φ).

When n= 1, the class of elliptical distributions coincides with the class of one-dimensional symmetric distributions. If X _∈ En(µ, I, φ) where I

is the identity matrix, then X is spherically distributed. Also, because of Proposition 1, every affinely transformed spherical random vector is ellip- tically distributed. The following stochastic representation shows that the converse is true if the transformation matrix has full rank.

Theorem 3. X _∈ En(µ,Σ, φ) with rank(Σ) = k if and only if there

exist a random variable R _≥ 0 independent of U, a k-dimensional random vector uniformly distributed on the unit sphere Sk_{, and a n×k matrix Λ}

withΛΛT = Σ, such that

X =d µ+RΛU

where =d denotes equality in distribution. Proof. For the proof, see Fang et al. (1994).

If X _∈ En(µ,Σ, φ), where Σ is a diagonal matrix, then X has uncorre-

lated components if the variance of the components ofX is finite. If X has independent components, then X ∈ N(µ,Σ). It should be remarked that the multivariate normal distribution is the only one among the elliptical dis- tributions where uncorrelated components imply independent components. A random vector X ∈ En(µ,Σ, φ) does not necessarily have density. If X

has a density functionf(x), then it has a special form.

Theorem 4. Let X _∈ En(µ,Σ, φ) where µ ∈ Rn and Σ ∈ Rn×n is

positive definite. ThenX can be represented stochastically byX=d µ+RΛU

with ΛΛT = Σ according to Theorem 3. Further let the c.d.f. of R be absolutely continuous. Then the p.d.f. of X is given by

fX(x) = p det(Σ−1_)·g R (x−µ)TΣ−1(x−µ), x6=µ, (1.27) where gR(t) = Γ n₂ 2πn/2t− n−1 2 ·f R _√ t, t >0 (1.28) andfR(t) is the p.d.f. of R.

Proof. A proof and a more general result for semi-definite Σ can be found in Frahm (2004).

The function gR(·) is called density generator. Given the density of the

generating variate R, one can compute the density generator of the corre- sponding elliptical distribution. Note that the contour lines of the density function form ellipsoids in Rn_{. For this reason the elliptical distributions}

are often calledelliptically contoured distributions.

Given the distribution ofX, the representationEn(µ,Σ, φ) is non-unique.

It uniquely determinesµbut Σ andφ(_·) are determined up to a positive con- stant. More precisely, ifX_∈En(µ1,Σ1, φ1) and X∈En(µ2,Σ2, φ2), then

µ1=µ2 Σ1 =cΣ2 φ1(·) =φ2(·/c)

for some constant c > 0. It comes out that it is possible to choose the characteristic generator φsuch that cov(X) = Σ if covariances are defined, see Embrechts et al. (2003) for an example.

Affine transformations of elliptical random vectors have also elliptical distribution

Theorem 5. Let X_∈En(µ,Σ, φ) andB be a q×n matrix and b∈Rq.

Then

Proof. For a proof, see Embrechts et al. (2003). If we partition X,µand Σ into

X= X1 X2 µ= µ1 µ2 Σ = Σ11 Σ12 Σ21 Σ22

whereX1 andµ1 arer×1 vectors and Σ11 is ar×r matrix. As a corollary from the above theorem if follows that

X1 ∈Er(µ1,Σ11, φ) and X2∈En−r(µ2,Σ22, φ)

Therefore the marginal distributions of the elliptical distributions are ellip- tical and with the same characteristic generator. The next result states that the conditional distribution of X1 given X2 is elliptical but in general not with the same characteristic generator.

Theorem 6. LetX ∈En(µ,Σ, φ)withΣstrictly positive definite. Then

the conditional distribution of X1 given that X2 =x X1|X2 =x∈Er(˜µ,Σ,˜ φ)˜

whereµ˜=µ1+ Σ12Σ−₂₂1(x−µ2)and Σ = Σ11˜ −Σ12Σ−₂₂1Σ21. Moreover φ˜=φ

if and only if X∈N(µ,Σ).

Proof. For the proof, see Fang et al. (1994).

The next result states that linear combinations of independent, ellipti- cally distributed random vectors with the same dispersion matrix Σ up to a positive constant remain elliptical.

Theorem 7. Let X _∈ En(µ,Σ, φ) and Xe ∈En(µ, cΣ,e φ)e with c > 0 be

independent. Then for a, b_∈R_,

aX+bXe _∈En(aµ+bµ,e Σ, φ∗), where φ∗(u) =φ(a2u)φ(be 2cu)

Proof. The proof uses the partficular form of the characteristic function from the definition, see Embrechts et al. (2003).

A practical problem with the elliptical distributions in multivariate risk modeling is that all marginals are of the same type, which may not be very realistic. In the remaining part of the section we give as examples some of the most widely used representatives of the class.

Example 1. (Multivariate Gaussian distribution) Let µ∈Rn _and

Λ _∈ Rn×n _{such that Σ := ΛΛ}T _{∈ R}n×n _{is positive definite. The random}

vector X _∈N(µ,Σ) is elliptically distributed since it is representable as

X=d µ+pχ2

where U is a n-dimensional random vector with uniform distribution on the unit sphere Sn _{and χ}2

n is a χ2-distributed random variable with n degrees

of freedom, independent ofU, see Frahm (2004) and the references therein. The random variable qχ2

k is the generating variate of X. This is easily

seen by considering the standard normal distribution which is the underlying spherical distribution with characteristic generatorφX(s) = exp(−s/2). The

density generator can be readily obtained becauseR=pχ2

n,

gR(t) =

1

(2π)n/2 ·exp(−t/2)

and in line with Theorem 4, we obtain the multivariate Gaussian p.d.f.

fX(x) = √ det Σ−1 (2π)n/2 exp −(x−µ) T_Σ−1_(x−µ) 2 , x_∈Rn _(1.29)

Example 2. (Multivariate t-distribution) Consider the random vec- tor Y =d µ+qX χ2 ν ν , ν ∈N

where µ _∈ Rn _{and X ∈ N(0,Σ). Then Y is said to be multivariate t-}

distributed withν degrees of freedom, location vectorµand dispersion matrix

Σ, Y ∈tn(µ,Σ, ν). The random vector allows for the stochastic representa-

tion Y =d µ+_q1 χ2 ν ν ·pχ2 nΛU where χ2

ν and χ2n are independent and have χ2 distribution, U is uniformly

distributed on the unit sphere Sn _{provided that Λ has full rank, and is in-}

dependent of χ2_ν and χ2_n, Σ = ΛΛT. Hence for the generating variate we have R=d s χ2 n χ2 ν ν d =pnFn,ν

where Fn,ν denotes a F-distributed random variable. The density generator

is gR(t) = Γ n+₂ν Γ ν₂ · 1 (νπ)n/2 · 1 + t ν −n+ν 2

fX(x) = Γ n+₂ν Γ ν₂ · √ det Σ−1 (νπ)n/2 · 1 +(x−µ) T_Σ−1_(x−µ) ν −n+ν 2 , x_∈Rn (1.30)