Modelling Dependence with Copulas - IFOR

8 Mixture of Extremal Distributions
Let G n j ,j =1,... ,2n−1 denote the extremal distributions with margins F 1 ,... ,F n
and rank correlation matrices ρ n j . Convex combinations
2 n−1
G n λ = ∑
j=1
2
λ j G n j ,λ ∑
n−1
j ≥ 0,
j=1
λ j =1, (8.0.7)
have the same marginals and rank correlation matrix given by ρ n λ = ∑ 2 n−1
j=1 λ jρ n j .
The subscript λ is to indicate that the chosen coefficients in the convex combination
is relevant.
The problem we will study in this chapter is the following:
Given marginal distributions F 1 ,... ,F n and an n × n rank
correlation matrix ρ, are there coefficients λ j such that G n λ
given above has those margins and rank correlation matrix?
Since all choices of coefficients λ j in the convex combination will result in G n λ having
the prescribed margins, the problem only lies in whether there exists λ j :s such that
ρ has a convex decomposition in the class of extremal rank correlation matrices
ρ j ,j =1,... ,2 n−1 . Thus the problem can be restated:
Given an arbitrary n × n rank correlation matrix ρ, is
there an n-vector λ such that
ρ =
2∑
n−1
j=1
2
λ j ρ n j ,λ ∑
n−1
j ≥ 0,
This can be formulated in an alternative way:
j=1
λ j =1 ?
Given an arbitrary n × n rank correlation matrix ρ, is
there a non-negative solution to the equation system
2∑
n−1
j=1
2 n−1
λ j ρ n j − ρ =0, ∑
j=1
λ j − 1 = 0 ? (8.0.8)
However, there exist rank correlation matrices that are not decomposable in the
class of extremal rank correlation matrices. And if such a decomposition exists, it
is in general not unique. Examples of such matrices are easy to find.
Example 8.1. Consider the linear correlation matrix ρ given by
⎛
⎞
1 0.3 0.2 0.5
ρ = ⎜ 0.3 1 0.4 0.7
⎟
⎝ 0.2 0.4 1 0.8 ⎠ .
0.5 0.7 0.8 1
It is easily verified that ρ is positive-definite, and hence a proper linear correlation
matrix. By applying the transformation
f : ρ ij ↦→ 6 π arcsin ρ ij
2 ,
to all elements in ρ, f(ρ) is a proper Spearman’s rank correlation matrix for some
Gaussian copula. We now look for a nonnegative solution to the equation system
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