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Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors. (English) Zbl 1409.62218

Summary: In this work, we focus on some conditional extreme risk measures estimation for elliptical random vectors. In a previous paper, we proposed a methodology to approximate extreme quantiles, based on two extremal parameters. We thus propose some estimators for these parameters, and study their consistency and asymptotic normality in the case of heavy-tailed distributions. Thereafter, from these parameters, we construct extreme conditional quantiles estimators, and give some conditions that ensure consistency and asymptotic normality. Using recent results on the asymptotic relationship between quantiles and other risk measures, we deduce estimators for extreme conditional \(L_{p}\)-quantiles and Haezendonck-Goovaerts risk measures. Under similar conditions, consistency and asymptotic normality are provided. In order to test the effectiveness of our estimators, we propose a simulation study. A financial data example is also proposed.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62H12 Estimation in multivariate analysis
60E05 Probability distributions: general theory
62G32 Statistics of extreme values; tail inference
62G08 Nonparametric regression and quantile regression
91G70 Statistical methods; risk measures

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References:

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