Definitions of tail dependence for multivariate random vectors are mostly related to their bivariate marginal distribution functions. Loosely speaking, tail dependence describes the limiting proportion that one margin exceeds a certain threshold given that the other margin has already exceeded that threshold. The following approach, as provided in the monograph of Joe (1997), represents one of many possible definitions of tail dependence.
Let
be a two-dimensional random vector. We say
that
is (bivariate) upper tail-dependent if:
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(3.3) |
Figures 3.1 and 3.2 illustrate tail dependence for a
bivariate normal and -distribution. Irrespectively of the correlation coefficient
the bivariate normal distribution is (upper) tail independent. In contrast, the bivariate
-distribution
exhibits (upper) tail dependence and the degree of tail dependence is affected by the correlation coefficient
The concept of tail dependence can be embedded within the copula theory. An
-dimensional distribution function
is called a copula if it has one-dimensional margins which
are uniformly distributed on the interval
Copulae are
functions that join or ``couple'' an
-dimensional distribution
function
to its corresponding one-dimensional
marginal distribution functions
, in the
following way: