When the POT-approach Fails: Super Heavy Tailed Distributions and Covariate Information

We consider the case that the conditional distribution of a random variable Y given X = x is in the domain of attraction of a generalized Pareto distribution (GPD), which parameters depend on x. In many cases this entails that the distribution of Y is not in the domain of attraction of any GPD anymore. We distinguish two cases whether the covariate variable X can be observed or not. Considering the first case we propose a conditional point process model to estimate the tail of the conditional distribution via Maximum-Likelihood. In the second case we derive limiting distribution of exceedances from super-heavy tailed distributions using nonlinear transformations.
Finally some open questions are addressed concerning threshold selection in the first case and discriminating between heavy and super-heavy tailed distributions in the second case.