Improved reduced-bias tail index and quantile estimators

• Published in: Journal of statistical planning and inference 2008-07, Vol.138 (6), p.1851-1870
• Main Authors: Beirlant, Jan, Figueiredo, Fernanda, Ivette Gomes, M., Vandewalle, Björn
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Extreme value statistics; General topics; Heavy tails; High quantiles; Linear inference, regression; Mathematics; Nonparametric inference; Parametric inference; Probability and statistics; Sciences and techniques of general use; Statistics
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2007.07.015

In this paper, we deal with bias reduction techniques for heavy tails, trying to improve mainly upon the performance of classical high quantile estimators. High quantiles depend strongly on the tail index γ , for which new classes of reduced-bias estimators have recently been introduced, where the second-order parameters in the bias are estimated at a level k 1 of a larger order than the level k at which the tail index is estimated. Doing this, it was seen that the asymptotic variance of the new estimators could be kept equal to the one of the popular Hill estimators. In a similar way, we now introduce new classes of tail index and associated high quantile estimators, with an asymptotic mean squared error smaller than that of the classical ones for all k in a large class of heavy-tailed models. We derive their asymptotic distributional properties and compare them with those of alternative estimators. Next to that, an illustration of the finite sample behavior of the estimators is also provided through a Monte Carlo simulation study and the application to a set of real data in the field of insurance.