Archimedean copulae form an important class of copulae
which are easy to construct and have good analytical properties.
A bivariate Archimedean copula has the form
for some continuous,
strictly decreasing, and convex generator function
such that
and the
pseudo-inverse function
is defined by:
We call strict if
In that case
Within the framework of tail
dependence for Archimedean copulae, the following result can be shown
(Schmidt; 2003).
Note that the one-sided derivatives of
exist, as
is a convex function.
In particular,
and
denote the one-sided derivatives at the
boundary of the domain of
Then:
Tables 3.1 and 3.2 list various Archimedean copulae in the same ordering as provided in Nelsen (1999, Table 4.1, p. 94) and in Härdle, Kleinow, and Stahl (2002, Table 2.1, p. 42) and the corresponding upper and lower tail-dependence coefficients (TDCs).
Number & Type | ![]() |
Parameters |
(1)
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(2) |
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(3)
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(4)
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(12) |
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(14) |
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(19) |
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In this section, we calculate the tail-dependence coefficient for
elliptically-contoured distributions (briefly: elliptical
distributions). Well-known elliptical distributions are the
multivariate normal distribution, the multivariate
-distribution, the multivariate logistic distribution, the multivariate symmetric stable distribution,
and the multivariate symmetric generalized-hyperbolic distribution.
Elliptical distributions are defined as follows: Let be an
-dimensional random vector and
be a
symmetric positive semi-definite matrix. If
, for some
possesses a characteristic function of the form
for some function
, then
is said to be elliptically
distributed with parameters
(location),
(dispersion), and
Let
denote the
class of elliptically-contoured distributions with the latter
parameters. We call
the characteristic generator. The
density function, if it exists, of an elliptically-contoured
distribution has the following form:
for some function
which we call the
density generator.
Observe that the name ``elliptically-contoured distribution'' is related to the elliptical contours of the latter density. For a more detailed treatment of elliptical distributions see the monograph of Fang, Kotz, and Ng (1990) or Cambanis, Huang, and Simon (1981).
In connection with financial applications, Bingham and Kiesel (2002) and Bingham, Kiesel, and Schmidt (2002) propose a
semi-parametric approach for elliptical distributions by estimating the
parametric component (
) separately from the
density generator
In their setting, the density generator is estimated by means of a nonparametric statistics.
Schmidt (2002b) shows that bivariate elliptically-contoured distributions are upper and lower tail-dependent if the tail of their density generator is regularly varying, i.e. the tail behaves asymptotically like a power function. Further, a necessary condition for tail dependence is given which is more general than regular variation of the latter tail: More precisely, the tail must be O-regularly varying (see Bingham, Goldie, and Teugels (1987) for the definition of O-regular variation). Although the equivalence of tail dependence and regularly-varying density generator has not been shown, all density generators of well-known elliptical distributions possess either a regularly-varying tail or a not O-regularly-varying tail. This justifies a restriction to the class of elliptical distributions with regularly-varying density generator if tail dependence is required. In particular, tail dependence is solely determined by the tail behavior of the density generator (except for completely correlated random variables which are always tail dependent).
The following closed-form expression exists (Schmidt; 2002b) for the
upper and lower tail-dependence coefficient of an
elliptically-contoured random vector
with positive-definite matrix
Note that corresponds to the
``correlation'' coefficient when it exists (Fang, Kotz, and Ng; 1990).
Moreover, the upper tail-dependence coefficient
coincides with the lower tail-dependence coefficient
and depends only on the ``correlation'' coefficient
and the
regular variation index
see Figure 3.3.
Table 3.3 lists various elliptical distributions,
the corresponding density generators (here denotes a
normalizing constant depending only on the dimension
) and the
associated regular variation index
from which one easily
derives the tail-dependence coefficient using formula
(3.6).
For many other closed form copulae one can explicitly derive the tail-dependence coefficient. Tables 3.4 and 3.5 list some well-known copula functions and the corresponding lower and upper TDCs.
Number & Type
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Parameters |
(28) Raftery |
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(29) BB1 |
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(30) BB4 |
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(31) BB7 |
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(32) BB8 |
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(33) BB11 |
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(34)
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Number & Type
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Parameters | upper-TDC | lower-TDC |
(28) Raftery |
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0 |
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(29) BB1 |
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(30) BB4 |
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(31) BB7 |
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(32) BB8 |
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0 |
(33) BB11 |
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(34)
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