Bivariate Extreme Value Distributions

The title of this chapter is deliberately chosen to focus on the bivariate case instead of the general multivariate. The reason is mainly one of expediency, because the general multivariate case would easily embroil us in the necessity to roll out a heavy machinery of notation without contributing to a deeper understanding of the issues involved. For a discussion of the general multivariate case, the reader may consult the book by Beirlant et al. (Statistics of Extremes. Chichester: John Wiley & Sons, Ltd.; 2004) The extension of extreme value statistics from the univariate to the multivariate case meets with several challenges. First of all, there is no direct simple generalization of the univariate extreme value types theorem to the multivariate case, and in particular, this also applies to the bivariate case. It is therefore of considerable interest to note that the ACER method can easily be extended to several dimensions, in particular, to two (Naess, A note on the bivariate ACER method. Preprint Statistics No. 01/2011, Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim; 2011). By this fact, a vehicle is obtained for providing a nonparametric statistical estimate of the bivariate extreme value distribution inherent in a bivariate time series. It will be seen that the bivariate ACER function is able to cover both spatial and temporal dependence characteristics of the given time series. Thus, it covers all simultaneous and non-simultaneous extreme events. From a practical point of view, this makes it possible to investigate the true behavior of the bivariate extreme value distribution for a particular case, and at the same time check the validity of the proposed copula models for bivariate extremes.