These two plots help us to understand the dependence relationship
between the two data set. The sign of \(\chi(u)\) determines
if the variables are positively or negatively correlated. Two variable
are asymptotically independent if \(\lim_{u\rightarrow1} \chi(u) =
0\). For the independent case, \(\chi(u) =
0\) for all u in (0,1). For the perfect dependence case,
\(\chi(u) = 1\) for all u in (0,1). Note that for a
bivariate extreme value model, \(\chi(u) = 2(1 - A(0.5))\) for all u in (0,1).
The measure \(\overline{\chi}\) is only useful for
asymptotically independent variables. Indeed, for asymptotically
dependent variable, we have \(\lim_{u\rightarrow
1}\overline{\chi}(u) = 1\). For
asymptotically independent variables, \(\lim_{u\rightarrow
1}\overline{\chi}(u)\) reflects the strength
of the dependence between variables. For independent variables,
\(\overline{\chi}(u) = 0\) for all u in (0,1).
If there is (short range) dependence between observations, users may
need to use bootstrap confidence intervals. Bootstrap series are
obtained by sampling contiguous blocks, of length l say,
uniformly with replacement from the original observations. The block
length l should be chosen to be much greater than the
short-range dependence and much smaller than the total number of
observations.