Modeling and estimation of conditional excesses

We investigate conditions for the existence of the limiting distribution of a bivariate random vector when one component becomes large. We revisit the existing literature on the topic, and present some new sufficient conditions. We focus on conditions which can be expressed in terms of geometric properties of the level curves of the density of the random vector. When the limiting conditional distribution exists, we propose consistent nonparametric estimators of this distribution and of the related normalizing sequences. We also provide a semi-parametric extrapolation procedure that allows the estimation of conditional probabilities when the conditioning event is extreme. A small simulation study illustrates our results.