Modelling Dependence with Copulas - IFOR

2.3 The Fréchet-Hoeffding Bounds for Joint Distribution Functions
Definition 6. Let F be a distribution function. Then the quasi-inverse of F is any
function F (−1) with domain I such that
1. if t is in RanF ,thenF (−1) (t) is any number x in R such that F (x) =t
i.e., for all t in RanF , F (F (−1) (t)) = t;
2. if t is not in RanF ,then
F (−1) (t) = inf{x|F (x) ≥ t} =sup{x|F (x) ≤ t}.
If F is strictly increasing, then the quasi-inverse is the ordinary inverse, which
we denote F −1 .
Corollary 2.1. Let H, C, F 1 ,F 2 ,... ,F n be as in Theorem 2.2, and let
F (−1)
1 ,F (−1)
2 ,... ,F n
(−1) be quasi-inverses of F 1 ,F 2 ,... ,F n , respectively. Then for
any u in I n ,
C(u 1 ,u 2 ,... ,u n )=H(F (−1)
1 (u 1 ),F (−1)
2 (u 2 ),... ,F n (−1) (u n )). (2.2.2)
Example 2.1. Let Φ denote the standard univariate normal distribution function
and let Φ n ρ denote the standard multivariate normal distribution function with linear
correlation matrix ρ. Then
C(u 1 ,u 2 ,... ,u n )=Φ n ρ (Φ−1 (u 1 ), Φ −1 (u 2 ),... ,Φ −1 (u n ))
is the Gaussian or normal n-copula. This copula is used very often in practice
because it has some nice properties as a member of the family of elliptical copulas,
and as it is easy to simulate from as well as practitioners knowledge of copulas is
quite limited. That the linear correlation matrix is used as parameterization of the
copula is not standard for copulas. The reasons for this follow from results in the
preceding chapters.
2.3 The Fréchet-Hoeffding Bounds for Joint Distribution
Functions
Consider the functions M n ,Π n and W n given by
M n (u) = min(u 1 ,u 2 ,... ,u n );
Π n (u) = u 1 u 2 ...u n ;
W n (u) = max(u 1 + u 2 + ...+ u n − n +1, 0).
The functions M n and Π n are n-copulas for all n ≥ 2 whereas the function W n is
not a copula for any n>2 as shown in the following example.
Example 2.2. Consider the n-cube [1/2, 1] n ⊂ I n . Because W is symmetric the
W -volume of [1/2, 1] n is given by
V W ([1/2, 1] n ) = max(1+...+1− n +1, 0) −
−n max(1/2+1+...+1− n +1, 0) +
( n
+ max(1/2+1/2+1+...+1− n +1, 0) −
2)
...
+max(1/2+...+1/2 − n +1, 0) =
= 1− n (1/2) + 0 + ...+0.
Hence, W n is not a copula for n ≥ 3.
The following theorem is the n-dimensional version of the Fréchet-Hoeffding
bounds inequality.
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