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

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 methods
• Semi-parametric estimation
• Statistics of extremes
• Value at risk

Mathematics Subject Classification:

• Primary 62G32, 62E20
• Secondary 65C05

Acknowledgment

Research partially supported by FCT/POCI and PPDCT/FEDER.