A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator

Abstract

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 (VaR p ), 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 VaR p 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.

EVT-based estimation of risk capital and convergence of high quantiles

“…Models encountered in practice often correspond to the case in which ρ = 0. Especially in the latter case, more research on the statistical estimation of high quantiles using EVT is needed; see, for instance, [17] and the references therein for some ideas.…”

Conditional heavy-tail behavior with applications to precipitation and river flow extremes

“…Comparison of L-estimatorsγ L p1q of γp1q " η 0`η1 based on 1000 independent samples of DGP (12) with µ " p2, 0q 1 , σ " p1, 0q 1 , η " p0.4, η 1 q and sample length n " 500 for different selection rules of k: pLq k " t2n 2{3 u, pL˚q k " k˚, pL˚2q k " t0.75¨k˚u and pL˚3q k " t1.25¨k˚u, where k˚is a data-adaptive rule defined in ( the asymptotically optimal choice for independent and identically GEV-distributed observations [Gomes and Pestana, 2007] and indeed, this choice led to the best results in our simple scenarios with GEV innovations. In practice, however, we may not always expect that the observations stem from a known parametric family and it may be preferable to choose a data adaptive rule.…”

A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator

“…For the use of this estimator in quantile estimation, see Gomes and Pestana (2007) and Caeiro and Gomes (2008b).…”

“…We merely need to have adequate ways of estimating the optimal level k 1 = k S1. Given a sample (X 1 , X 2 , · · · , X n ), plot, for τ = 0 and τ = 1, the estimatesρ τ (k) in (17).…”