Title: Estimation of conditional extreme risk measures from heavy-tailed elliptical random vector
Authors: Antoine Usseglio-Carleve - Institut Camille Jordan (France) [presenting]
Abstract: In recent years, the question of estimating extreme quantiles, or more generally extreme risk measures has seen many advances. We consider an elliptical random vector, and focus on the extreme quantiles of a component conditioned by all the others. For that purpose, we start by recalling the main properties of elliptical distributions, especially the consistency property. Then we introduce a heavy-tail assumption on the marginal distributions. Once the frame has been defined, we propose in a first time an asymptotic relationship between conditional and unconditional quantiles, based on two parameters, called extremal parameters. Under our assumption, we easily provide estimators for both extremals parameters, and give their asymptotic distribution. Then, using the regular variation properties induced by our assumption, we deduce estimators for extreme conditional quantiles, and give some simple conditions for consistency. On the other hand, a stronger assumption and other conditions are required for asymptotic normality. A simulation study with a Student vector is proposed, in order to compare our estimators with theoretical results. The choice of the sequences for the tail index and kernel estimators and quantile level is also discussed through boxplots. We conclude by showing that many extreme risk measures may be deduced from extreme quantiles, like Haezendonck-Goovaerts risk measures, or Lp-quantiles. A financial data example is finally proposed.