Title: Threshold Selection for Multivariate Heavy-Tailed Data,
Abstract: Regular variation is often used as the starting point for modeling
multivariate heavy-tailed data. A random vector is regularly varying if and
only if its radial part $R$ is regularly varying and is asymptotically
independent of the angular part $\Theta$ as $R$ goes to infinity. The
conditional limiting distribution of $\Theta$ given $R$ is large characterizes
the tail dependence of the random vector and hence its estimation is the
primary goal of applications. A typical strategy is to look at the angular
components of the data for which the radial parts exceed some threshold. While
a large class of methods has been proposed to model the angular distribution
from these exceedances, the choice of threshold has been scarcely discussed in
the literature. In this paper, we describe a procedure for choosing the
threshold by formally testing the independence of $R$ and $\Theta$ using a
measure of dependence called distance covariance. We generalize the limit
theorem for distance covariance to our unique setting and propose an algorithm
which selects the threshold for $R$. This algorithm incorporates a subsampling
scheme that is also applicable to weakly dependent data. Moreover, it avoids
the heavy computation in the calculation of the distance covariance, a typical
limitation for this measure. The performance of our method is illustrated on
both simulated and real data.