In this section we investigate the finite-sample properties of
the introduced TDC estimators. One thousand independent copies of
and
i.i.d. random vectors (
denotes the sample length) of a bivariate standard
-distribution with
and
degrees of freedom are
generated and the upper TDCs are estimated. Note that the parameter
equals the
regular variation index
which we discussed in the context of elliptically-contoured distributions.
The empirical bias
and root-mean-squared error (RMSE) for all three introduced TDC
estimation methods are derived and presented in Tables
3.6, 3.7, and
3.8, respectively.
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Regarding the parametric
approach we apply the procedure introduced in Section
3.4 and estimate by a trimmed
empirical correlation coefficient with trimming proportion
and
by a Hill estimator. For the latter we choose the optimal
threshold value
according to Drees and Kaufmann (1998).
The empirical bias and RMSE corresponding to the estimation of
and
are provided in
Tables 3.9 and
3.10.
Observe that Pearson's correlation coefficient
does not exist
for
In this case,
denotes some dependence
parameter and a more robust estimation procedure should be used
(Frahm et al.; 2002).
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Finally, Figures 3.4 and 3.5 illustrate the (non-)parametric
estimation results of the upper TDC estimator
Presented are
TDC
estimations with sample lengths
and
The plots visualize the decreasing empirical bias and variance for increasing sample length.
The empirical study shows that the TDC estimator
outperforms the other two estimators. For
the bias (RMSE) of
is three (two and
a half)
times larger than the bias (RMSE) of
,
whereas the bias (RMSE) of
is two (ten percent) times larger than the bias (RMSE) of
. More empirical and statistical results
regarding the estimators
and
are given in Schmidt and Stadtmüller (2003).
However, note that the estimator
was especially
developed for bivariate elliptically-contoured distributions. Thus,
the estimator
is recommended for TDC
estimations of non-elliptical or unknown bivariate distributions.