Suppose
are i.i.d. bivariate random
vectors with distribution function
and copula
We assume
continuous marginal distribution functions
Tests for tail dependence or tail independence are
given for example in Ledford and Tawn (1996) or Draisma et al. (2004).
We consider the following three (non-)parametric estimators for the
lower and upper tail-dependence coefficients and
These estimators have been discussed in Huang (1992) and Schmidt and Stadtmüller (2003).
Let
be the empirical copula defined by:
with and
denoting the empirical distribution functions
corresponding to
and
respectively. Let
and
be the rank of
and
respectively. The first estimators are
based on formulas (3.1) and
(3.2):
and
where
and
as
and the first expression in
(3.8) has to be understood as the empirical
copula-measure of the interval
The
second type of estimator is already well known in multivariate extreme-value
theory (Huang; 1992). We only provide the estimator for the upper TDC.
with
and
as
The optimal choice of
is related to the usual
variance-bias problem and we refer the reader to Peng (1998) for more
details.
Strong consistency and asymptotic normality for both types of nonparametric estimators are also addressed in
the latter three reference.
Now we focus on an elliptically-contoured bivariate random vector
In the presence of tail dependence, previous arguments justify a sole consideration of elliptical
distributions having a regularly-varying density generator with
regular variation index
This
implies that the distribution function of
has also a
regularly-varying tail with index
. Formula (3.6)
shows that the upper and lower tail-dependence coefficients
and
depend only on the regular variation index
and the ``correlation'' coefficient
Hence, we propose the
following parametric estimator for
and
:
Several robust estimators
for
are provided
in the literature such as estimators based on techniques of multivariate trimming (Hahn, Mason, and Weiner; 1991),
minimum-volume ellipsoid estimators (Rousseeuw and van Zomeren; 1990), and
least square estimators (Frahm et al.; 2002).
For more details regarding the relationship between the regular variation index the density
generator, and the random variable
we refer to
Schmidt (2002b). Observe that even though the estimator for the regular variation index
might be unbiased,
the TDC estimator
is biased due to the integral transform.