A Sturdy Reduced-Bias Extreme Quantile (<i>VaR</i>) Estimator

TLDR

Extreme‑value statistics aim to estimate high quantiles such as value‑at‑risk, but classical semiparametric tail estimators suffer from high bias when few extreme observations are used. This paper develops a bias‑reduced semiparametric estimator of the high‑quantile VaR for heavy‑.

Abstract

Abstract Please see the supplementary material for an important correction.The main objective of statistics of extremes lies in the estimation of quantities related to extreme events. In many areas of application, such as statistical quality control, insurance, and finance, a typical requirement is to estimate a high quantile, that is, the value at risk at a level p (VaRp), high enough so that the chance of an exceedance of that value is equal to p, small. In this article we deal with the semiparametric estimation of VaRp for heavy tails. The classical semiparametric estimators of parameters characterizing the tail behavior of the underlying model F usually exhibit a high bias for low thresholds, that is, for large values of k, the number of top order statistics used for the estimation. We shall here deal with bias reduction techniques for heavy tails, trying to improve the performance of the classical high quantile estimators through the use of an adequate bias-corrected tail index estimator. The new high quantile estimators have a mean squared error smaller than that of the classical estimators, even for small values of k. They are, thus, alternatives to the classical estimators not only around optimal levels but also for other levels. The asymptotic distributional properties of the proposed classes of estimators are derived. The estimators are compared with alternative ones, not only asymptotically but also for finite samples, through Monte Carlo techniques. An application to the analysis of different datasets in the field of finance is also provided.KEY WORDS: Heavy tailsHigh quantilesSemiparametric estimationStatistics of extremesValue at riskView correction statement:Corrections