Modelling Dependence with Copulas - IFOR

2 Copulas
Here C α1(X 1),α 2(X 2) is the survival copula, Ĉ, ofX 1 and X 2 , i.e.,
H(x 1 ,x 2 )=P[X 1 >x 1 ,X 2 >x 2 ]=Ĉ(F 1(x 1 ), F 2 (x 2 )).
Consider the trivariate case.
Let α 1 , α 2 and α 3 be strictly decreasing. Then
C α1(X 1),α 2(X 2),α 3(X 3)(u 1 ,u 2 ,u 3 )=
= C α2(X 2),α 3(X 3)(u 2 ,u 3 ) − C X1,α 2(X 2),α 3(X 3)(1 − u 1 ,u 2 ,u 3 )
= ...
= u 1 + u 2 + u 3 − 2+C X1,X 2
(1 − u 1 , 1 − u 2 )+C X1,X 3
(1 − u 1 , 1 − u 3 )
+C X2,X 3
(1 − u 2 , 1 − u 3 ) − C X1,X 2,X 3
(1 − u 1 , 1 − u 2 , 1 − u 3 ).
Here C α1(X 1),α 2(X 2),α 3(X 3) is the survival copula of X 1 ,X 2 ,X 3 .
Note also that the joint survival function of n uniform (0, 1) random variables
whose joint distribution function is the copula C is
C(u 1 ,u 2 ,... ,u n )=Ĉ(1 − u 1, 1 − u 2 ,... ,1 − u n ).
Although most results could be presented without needing measure theory,
there are results in the remaining sections that are best presented and understood
using some terminology and results from measure theory. Each joint distribution
H induces a probability measure on R n via V H ((−∞,x 1 ] × (−∞,x 2 ] ×
...× (−∞,x n ]) = H(x 1 ,x 2 ,... ,x n ). Since copulas are joint distribution functions,
each copula induces a probability measure on I n via V C ([0,u 1 ]×[0,u 2 ] ...×[0,u n ]) =
C(u 1 ,u 2 ,... ,u n ). Hence, at an intuitive level, the C-measure of a subset of I n is
the probability that n uniform (0, 1) random variables with joint distribution C
assume values in that subset.
For any copula C, let
where
C(u 1 ,u 2 ,... ,u n )=A C (u 1 ,u 2 ,... ,u n )+S C (u 1 ,u 2 ,... ,u n ),
∫ u1
∫ un
A C (u 1 ,... ,u n ) =
∂ n
...
C(u 1 ,... ,u n )ds 1 ... ds n ,
0 0 ∂u 1 ...∂u n
S C (u 1 ,... ,u n ) = C(u 1 ,... ,u n ) − A C (u 1 ,... ,u n ).
Unlike multivariate distributions in general, the margins of a copula are continuous,
hence a copula has no individual points in I n whose C-measure is positive.
If C = A C on I n then C is said to be absolutely continuous. In this case C has
∂
density
n
∂u 1...∂u n
C(u 1 ,... ,u n ).
If C = S C on I n ∂
then C is said to be singular, and n
∂u 1...∂u n
C(u 1 ,... ,u n )=0
almost everywhere in I n .
The support of a copula is the complement of the union of all open subsets of
I n with C-measure 0. When the support of a copula C is I n , it is said to have “full
support”. When C is singular its support has Lebesgue measure zero and conversely.
However a copula can have full support without being absolutely continuous.
Example 2.4. Consider the bivariate Fréchet-Hoeffding upper bound M. Since
∂ 2
∂u∂v M(u, v) = 0 everywhere on I2 except on the main diagonal (which has Lebesgue
measure zero), and the M-measure of every rectangle in I 2 entirely above or below
the main diagonal is zero, M is singular. The support of M is the main diagonal of
I 2 as shown in figure 2.1.
Similarly the support of the bivariate Fréchet-Hoeffding lower bound W is the secondary
diagonal of I 2 as shown in figure 2.1.
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