Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology

Abstract

Abstract In this article, we deal with the estimation, under a semi-parametric framework, of a positive extreme value index γ, the primary parameter in Statistics of Extremes, and associated estimation of the Value at Risk (VaR) at a level p, the size of the loss occurred with a small probability p. We consider second-order minimum-variance reduced-bias (MVRB) estimators, and propose the use of bootstrap computer-intensive methods for the adaptive choice of thresholds, both for γ and Var p . Applications in the fields of insurance and finance, as well as a small-scale simulation study of the bootstrap adaptive estimators’ behaviour, are also provided. Keywords: Bootstrap methodsSemi-parametric estimationStatistics of extremesValue at riskMathematics Subject Classification: Primary 62G32, 62E20Secondary 65C05 Acknowledgment Research partially supported by FCT/POCI and PPDCT/FEDER.