Environmental data: multivariate Extreme Value Theory in practice.

Let $(X_t , Y_t)$ be a bivariate stationary time series in some environmental study. We are interested to estimate the failure probability defined as $P(X_t > x,Y_t > y)$, where $x$ and $y$ are high return levels. For the estimation of high return levels, we consider three methods from univariate extreme value theory, two of which deal with the extreme clusters. We further derive estimators for the bivariate failure probability, based on Draisma et al. (2004)'s approach and on Heffernan and Tawn (2004)'s approach. The comparison on different estimators is demonstrated via a simulation study. To the best of our knowledge, this is the first time that such a comparative study is performed. Finally, we apply the procedures to the real data set and the results are discussed.