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(17.1) |
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(17.2) |
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(17.3) |
A function that satisfies the first property is called
grounded. The third property describes the analog of a
non-decreasing, one-dimensional function. A function with this
property is thus called 2-increasing. An example of a
non-decreasing function is
max. It is easy to see
that for
,
the third property is not
satisfied; thus this function is not 2-increasing. The function
is, on the other hand, 2-increasing,
yet decreasing for
.
The notation ``copula'' becomes clear in the following theorem from Sklar (1959).
A particular example is the product copula : two
random variables
and
are independent if and only if
The product copula is given by:
Another important example of a copula is the Gaussian or normal copula:
Here denotes the bivariate normal density function with
correlation
and
represents the gaussian
marginal distribution. In the case
we obtain:
An important class of copulas is the Gumbel-Hougaard Family, Hutchinson and Lai (1990), Nelsen (1999). This class is parameterized by
For
we obtain the product copula:
. For
we obtain the
minimum copula:
Moreover the differentiability of the copulas can be shown.
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(17.11) |
To illustrate this theorem we consider the Gumbel-Hougaard copula
(16.9):
It is easy to see that
is a strictly monotone
increasing function of
. The inverse
is
therefore well defined. How do copulas behave under
transformations? The next theorem gives some information on this.