Modelling Dependence with Copulas - IFOR

2 Copulas
Theorem 2.3. If C ′ is any n-subcopula, then for every u in DomC ′ ,
W n (u) ≤ C ′ (u) ≤ M n (u). (2.3.1)
For the proof, see Nelsen (1999) [13].
Although the Fréchet-Hoeffding lower bound W n is never a copula for n>2,
the left-hand side in (2.3.1) is the best possible in the sense that for all n ≥ 3and
any u in I n , there is an n-copula C such that C(u) =W n (u).
Theorem 2.4. For any n ≥ 3 and any u in I n , there is an n-copula C (which
depends on u) such that
For the proof, see Nelsen (1999) [13].
C(u) =W n (u).
Definition 7. If C 1 and C 2 are copulas, we say that C 1 is smaller than C 2 (or
C 2 is larger than C 1 ), and write C 1 ≺ C 2 (or C 2 ≻ C 1 )ifC 1 (u 1 ,u 2 ,... ,u n ) ≤
C 2 (u 1 ,u 2 ,... ,u n ) for all u 1 ,u 2 ,... ,u n in I.
Thus the Fréchet-Hoeffding lower bound W n is smaller than every n-copula C
and the Fréchet-Hoeffding upper bound M n is larger than every n-copula C. This
partial ordering of the set of copulas is called a concordance ordering. It is a partial
ordering since not every pair of copulas is comparable. However many important
parametric families of copulas are totally ordered. We call the one-parameter family
{
Cθ
}
positively ordered if Cθ1 ≺ C θ2 whenever θ 1 ≤ θ 2 and negatively ordered if
C θ1 ≻ C θ2 whenever θ 1 ≤ θ 2 .
2.4 Copulas and Random Variables
Let X 1 ,X 2 ,... ,X n be continuous random variables with distribution functions
F 1 ,F 2 ,... ,F n , respectively, and joint distribution function H. Then we say that
X 1 ,X 2 ,... ,X n have copula C, whereC is given by (2.2.1).
Theorem 2.5. Let X 1 ,X 2 ,... ,X n be continuous random variables with copula C.
Then X 1 ,X 2 ,... ,X n are independent if and only if C =Π n .
One nice property of copulas is that for strictly monotone transformations of
the random variables copulas are either invariant, or change in certain simple ways.
Note that if the distribution function of a random variable X is continuous, and if α
is a strictly monotone function whose domain contains RanX, then the distribution
function of the random variable α(X) is also continuous.
Theorem 2.6. Let X 1 ,X 2 ,... ,X n be continuous random variables with copula C.
If α 1 ,α 2 ,... ,α n are strictly increasing on RanX 1 , RanX 2 ,... ,RanX n respectively,
then α 1 (X 1 ),α 2 (X 2 ),... ,α n (X n ) have copula C. Thus C is invariant under strictly
increasing transformations of X 1 ,X 2 ,... ,X n .
Proof. Let F 1 ,F 2 ,... ,F n denote the distribution functions of X 1 ,X 2 ,... ,X n respectively,
and let G 1 ,G 2 ,... ,G n denote the distribution functions of α 1 (X 1 ),
α 2 (X 2 ), ... ,α n (X n ) respectively. Let X 1 ,X 2 ,... ,X n have copula C, and let
α 1 (X 1 ), α 2 (X 2 ), ... ,α n (X n ) have copula C α . Since α k is strictly increasing for
each k, G k (x) =P[α k (X k ) ≤ x] =P[X k ≤ α −1
k
(x)] = F k(α −1
k
(x)) for any x in R.
C α (G 1 (x 1 ),... ,G n (x n )) = P[α 1 (X 1 ) ≤ x 1 ,... ,α n (X n ) ≤ x n ]
= P[X 1 ≤ α −1
1 (x 1),... ,X n ≤ α −1
n (x n)]
= C(F 1 (α −1
1 (x 1)),... ,F n (α −1
n (x n )))
= C(G 1 (x 1 ),... ,G n (x n )).
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