Modelling Dependence with Copulas - IFOR

8 Mixture of Extremal Distributions
Theorem 8.1. Let X, Y and Z be random variables with joint distribution function
H and continuous margins F 1 , F 2 and F 3 respectively.
1. If (X, Y ) and (Y,Z) are comonotonic then (X, Z) is also comonotonic and
H(x, y, z) =min{F 1 (x),F 2 (y),F 3 (z)}.
2. If (X, Y ) is comonotonic and (Y,Z) is countermonotonic then (X, Z) is
countermonotonic and H(x, y, z) =max{0, min{F 1 (x),F 2 (y)} + F 3 (z) − 1}.
3. If (X, Y ) and (Y,Z) are countermonotonic then (X, Z) is comonotonic and
H(x, y, z) =max{0, min{F 1 (x),F 3 (z)} + F 2 (y) − 1}.
For the proof, see Embrechts, McNeil and Straumann (1999) [3].
Let ρ max and ρ min denote the maximum and minimum attainable correlations
between two random variables with given univariate distribution functions.
Theorem 8.2. Let F 1 ,... ,F n , n ≥ 3, be continuous distribution functions and
let ρ be a proper correlation matrix satisfying the following conditions for all i ≠ j,
i ≠ k and j ≠ k:
1. ρ ij ∈{ρ min (F i ,F j ),ρ max (F i ,F j )},
2. If ρ ij = ρ max (F i ,F j ) and ρ ik = ρ max (F i ,F k ) then ρ jk = ρ max (F j ,F k ),
3. If ρ ij = ρ max (F i ,F j ) and ρ ik = ρ min (F i ,F k ) then ρ jk = ρ min (F j ,F k ),
4. If ρ ij = ρ min (F i ,F j ) and ρ ik = ρ min (F i ,F k ) then ρ jk = ρ max (F j ,F k ).
Then there exists a unique distribution with margins F 1 ,... ,F n and correlation
matrix ρ. This distribution is known as an extremal distribution. In R n there are
2 n−1 possible extremal distributions.
Proof. Without loss of generality suppose
{
ρmax (F
ρ 1j =
1 ,F j ), 2 ≤ j ≤ m ≤ n,
ρ min (F 1 ,F j ), m < j ≤ n,
for some 2 ≤ m ≤ n. The pairwise relationship of any two margins is determined
by their pairwise relationship to the first margin. The margins for which
ρ 1j takes a maximal value form an equivalence class, as do the margins for which
ρ 1j takes a minimal value. Hence (X 1 ,... ,X m ) must be pairwise comonotonic and
(X m+1 ,... ,X n ) must be pairwise comonotonic, but (X k ,X l )where2≤ k ≤ m and
m