Pál Rakonczai, Institute of Mathematics, Probability Theory and Statistics, Budapest



Abstract
In the field of environmental studies or financial
mathematics the simultaneous appearance of peaks are subject of major
interest: either the joint extremes of several gauges or stocks, or
extremes of the main phenomena and high values in their explanatory
variables e.g. flood peaks and high rainfall, peaks of oil and other
commodity stock prices may define these problems. Mathematical
statistics offers sophisticated tools for examining the dependence
structure of such extreme events. An important one of those providing
a way to study the dependence structure of high values separately from
the marginal distributions is the so-called copula approach nowadays
especially widely used in the aforesaid areas. To find the appropriate
copula model necessitates the elaboration of finely tuned
goodness-of-fit (GOF) statistics. Though there are known measures, but
more often then not they are effective only in the two dimensional
case, and they are not really suitable to the above-mentioned problem
of the dependence of the multivariate extremes.  We were confronted
with the need to work out more appropriate methods when studying river
flow series in Hungary. We were able to increase substantially the
sensitivity of GOF evaluation by introducing appropriate weights into
the test-statistics, and in the end it helped to choose the model that
provided reliable long term estimates for flood risk. The main
approaches are based on the multivariate probability integral
transformation of the joint distribution, which reduces the
multivariate problem to one dimension. As a finer assessment of fit of
the dependence structure we also build a test-statistic analogous to
the previous one but based on a non-parametric estimator of the copula
density. The applications of the methods are illustrated on financial
and river flow data.

Keywords: Copula models, Goodness-of-Fit, Probability Integral Transformation,
Kernel estimations, River flow series, Stock prices