Minimum-variance reduced-bias estimation of the extreme value index: A theoretical and empirical study

In extreme value (EV) analysis, the EV index (EVI), , is the primary parame- ter of extreme events. In this work, we consider positive, that is, we assume that F is heavy tailed. Classical tail parameters estimators, such as the Hill, the Moments, or the Weissman estimators, are usually asymptotically biased. Con- sequently, those estimators are quite sensitive to the number of upper order statistics used in the estimation. Minimum-variance reduced-bias (RB) estima- tors have enabled us to remove the dominant component of asymptotic bias without increasing the asymptotic variance of the new estimators. The purpose of this paper is to study a new minimum-variance RB estimator of the EVI. Under adequate conditions, we prove their nondegenerate asymptotic behavior. More- over, an asymptotic and empirical comparison with other minimum-variance RB estimators from the literature is also provided. Our results show that the proposed new estimator has the potential to be very useful in practice

Bias correction in multivariate extremes

The estimation of the extremal dependence structure is spoiled by the impact of the bias, which increases with the number of observations used for the estimation. Already known in the univariate setting, the bias correction procedure is studied in this paper under the multivariate framework. New families of estimators of the stable tail dependence function are obtained. They are asymptotically unbiased versions of the empirical estimator introduced by Huang [Statistics of bivariate extremes (1992) Erasmus Univ.]. Since the new estimators have a regular behavior with respect to the number of observations, it is possible to deduce aggregated versions so that the choice of the threshold is substantially simplified. An extensive simulation study is provided as well as an application on real data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1305 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Second-order refined peaks-over-threshold modelling for heavy-tailed distributions

Modelling excesses over a high threshold using the Pareto or generalized Pareto distribution (PD/GPD) is the most popular approach in extreme value statistics. This method typically requires high thresholds in order for the (G)PD to fit well and in such a case applies only to a small upper fraction of the data. The extension of the (G)PD proposed in this paper is able to describe the excess distribution for lower thresholds in case of heavy tailed distributions. This yields a statistical model that can be fitted to a larger portion of the data. Moreover, estimates of tail parameters display stability for a larger range of thresholds. Our findings are supported by asymptotic results, simulations and a case study.Comment: to appear in the Journal of Statistical Planning and Inferenc

A Bias-reduced Estimator for the Mean of a Heavy-tailed Distribution with an Infinite Second Moment

We use bias-reduced estimators of high quantiles, of heavy-tailed distributions, to introduce a new estimator of the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked, in a simulation study, by four of the most popular goodness-of-fit tests for different sample sizes. Moreover, we compare, in terms of bias and mean squared error, our estimator with Peng's estimator (Peng, 2001) and we evaluate the accuracy of some resulting confidence intervals.Comment: Submitte