A comparison of two bivariate extreme value distributions

There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, namely Gumbel mixed (GM) model and Gumbel logistic (GL) model. These two models have completely different structures and their dependence ranges are different. The product-moment correlation coefficient for the former is [0,2/3] and the latter is [0,1]. It is natural to ask which one is more appropriate for representing the joint probabilistic behavior of two correlated Gumbel-distributed variables. This study compares these two models by numerical experiments. The comparison is based on that: (i) if the two distribution models are identical, then the joint probability and the joint return period computed by the GM model should be the same as those by the GL model; and (ii) if a selected distribution is the true distribution from which sample data are drawn, then the probabilities computed by the theoretical model should provide a good fit to empirical ones. Comparison results indicate that in the range of correlation coefficient [0,2/3], both models provide identical joint probabilities and joint return periods, and both indicate a good fit to empirical probabilities; while for (2/3,1), only the Gumbel logistic model can be used.