Modelling Dependence with Copulas - IFOR

6 Elliptical Copulas
Example 6.2. For the standard multivariate normal distribution R ∼ √ χ 2 n . For
the standard multivariate t-distribution with ν degrees of freedom R satisfies R 2 /n ∼
F (n, ν), where F (n, ν) denotes the F-distribution with n and ν degrees of freedom.
A random vector X ∼ S n (φ) does not necessarily have a density. However it
seems reasonable for our purpose to only consider spherical (and later elliptical)
distributions that possess densities. In this case it follows from Theorem 6.1 that
the densities must be of the form g(x T x) for some nonnegative function g of one
scalar variable. Hence the contours of equal density form spheres in R n .
Definition 18. An n×1 random vector X is said to have an elliptically symmetric
distribution (or simply elliptical distribution) with parameters µ and Σ if
where A : n × k and AA T = Σ with rank(Σ) = k.
X = d µ + AY, Y ∼ S k (φ) (6.0.9)
It follows that the characteristic function can be written as
Ψ(t) = E(exp(it T X)) = E(exp(it T (AY + µ)))
= exp(it T µ)E(exp(i(A T t) T Y)) = exp(it T µ)φ((A T t) T (A T t))
= exp(it T µ)φ(t T Σt).
We write X ∼ EC n (µ, Σ,φ).
If Y has density g(y T y) and Σ is positive definite, then AY + µ has density
1
√
det(Σ)
g((x − µ) T Σ −1 (x − µ)),
so the contours of equal density form ellipsoids in R n .
If X ∼ EC n (µ, Σ,φ), where Σ is a diagonal matrix, then X has uncorrelated
components. If X has independent components, then X ∼N(µ, Σ). Note that the
multivariate normal distribution is the only one among the elliptical distributions
where uncorrelated components imply independent components.
Given the distribution of X, the representation EC n (µ, Σ,φ) is not unique. It
uniquely determines µ but Σ and φ are only determined up to a positive constant,
see Fang, Kotz and Ng (1987) [5]. We would like to have a representation such that
Cov(X) = Σ. Is there such a representation?
Cov(X) = Cov(µ + AY) =AA T Cov(Y)
= AA T E(R 2 )Cov(U),
provided E(R 2 ) < ∞.
Let Y ∼N n (0, I n ). Then we have Y = d ‖Y‖U, where‖Y‖ is independent of U.
Furthermore ‖Y‖ 2 ∼ χ 2 n,soE(‖Y‖ 2 )=n. Since Cov(Y) =I n we get the general
result:
If U is uniformly distributed on the unit hypersphere in R n ,thenCov(U) =I n /n.
Thus Cov(X) =AA T E(R 2 )I n /n. By choosing the characteristic generator φ ′ (s) =
φ(s/c), where c = E(R 2 )/n, wegetCov(X) = Σ. Hence an elliptical distribution
is fully described by its mean, it covariance matrix and its characteristic generator,
which can be chosen so that Cov(X) =Σ.
Theorem 6.2. If X ∼ EC m (µ, Σ,φ), B is an m × n matrix of rank m (m ≤ n)
and b is an m × 1 vector, then
b + BX ∼ EC m (b + Bµ,BΣB T ,φ).
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