CopulasΒΆ
In this part, we will define the concept of copula.
To define the joined probability density function of the random input
vector
by composition, one needs:
the specification of the copula of interest
with its parameters,
the specification of the
marginal laws of interest
of the
input variables
.
The joined cumulative density function is therefore defined by:
Copulas allow one to represent the part of the joined cumulative density
function which is not described by the marginal laws. It enables to
represent the dependency structure of the input variables. A copula is
a special cumulative density function defined on
whose marginal distributions are uniform on
. The choice
of the dependence structure is disconnected from the choice of the
marginal distributions.
A copula, restricted to
is a
-dimensional cumulative density function with uniform
marginals.
,
,
For all
-box
, we have
, where:
, the summation being made over the
vertices
of
.
if
for an even number of
,
otherwise.