Modelling Dependence with Copulas - IFOR

6 Elliptical Copulas
A spherical distribution distribution is an extension of the standard multinormal
distribution N (0, I n ) and an elliptical distribution is an extension of N (µ, Σ). Recall
that N (µ, Σ) can be defined from the standard multivariate normal distribution
N (0, I n )via
X = d µ + AY, (6.0.6)
where X ∼N(µ, Σ), Y ∼N(0, I n )andΣ=AA T . With this in mind it seems
natural to start with spherical distributions and then define elliptical distributions
from the spherical in the way indicated by (6.0.6).
Definition 16. An n × 1 random vector X is said to have a spherically symmetric
distribution (or simply spherical distribution) if for every Γ ∈O(n),
where O(n) denotes the set of n × n orthogonal matrices.
ΓX = d X, (6.0.7)
Note that an orthogonal matrix represents a rotation transformation. Hence
spherical distributions are geometrically interpreted as those invariant under rotations.
Theorem 6.1. An n-vector X has a spherical distributions if and only if its characteristic
function Ψ(t) satisfies one of the following equivalent conditions:
1. Ψ(Γ T t)=Ψ(t) for any Γ ∈O(n);
2. There exist a function φ(·) of a scalar variable such that
Ψ(t) =φ(t T t).
For the proof, see Fang, Kotz and Ng (1987) [5].
From the theorem above we see that for spherical distributions the characteristic
function Ψ(t) =E(exp(it T X)) = φ(t T t), and for this reason we will use the notation
X ∼ S n (φ) whenX is spherically distributed. The function φ given above is called
the characteristic generator of the spherical distribution.
Example 6.1. Let X ∼N(0, I n ). Since the components X i ∼N(0, 1),i=1,... ,n
are independent and the characteristic function of X i is exp(−t 2 i /2), the characteristic
function of X is
exp{− 1 2 (t2 1 + ...+ t 2 n)} =exp{− 1 2 tT t}.
From Theorem 6.1 it then follows that X ∼ S n (φ) whereφ(s) =exp(−s/2).
The spherical distributions can be defined in an alternative and equivalent way.
This definition also provides the basis for many random variate generation techniques.
Definition 17. An n × 1 random vector X is said to have a spherically symmetric
distribution (or simply spherical distribution) if it has the stochastic representation
X = d RU (6.0.8)
for some positive random variable R independent of the random vector U uniformly
distributed on the unit hypersphere S n−1 = {z ∈ R n |z T z =1}.
Spherical distributions can thus be interpretated as mixtures of uniform distributions
on hyperspheres with different radius.
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