Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2008-02, Vol.70 (1), p.31-52
• Main Authors: Ivette Gomes, M, de Haan, Laurens, Rodrigues, Lígia Henriques
• Format: Article
• Language: English
• Subjects: Asymptotic value; Bias; Confidence interval; Consistent estimators; Distribution; Estimation; Estimation bias; Estimators; Exact sciences and technology; General topics; Heavy tails; Linear inference, regression; Log-excesses; Mathematical expressions; Mathematics; Maximum likelihood; Maximum likelihood estimation; Maximum likelihood method; Monte Carlo simulation; Nonparametric inference; Parametric inference; Probability and statistics; Sampling bias; Sciences and techniques of general use; Semiparametric estimation; Statistical methods; Statistical variance; Statistics; Statistics of extremes; Studies; Traffic estimation
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1467-9868.2007.00620.x

We are interested in the derivation of the distributional properties of a weighted log-excesses estimator of a positive tail index γ. One of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias, together with the maintenance of the asymptotic variance of the maximum likelihood estimator of γ, under a strict Pareto model. We consider the external estimation not only of a second-order shape parameter ρ but also of a second-order scale parameter β. This will enable us to reduce the asymptotic variance of the final estimators under consideration, compared with second-order reduced bias estimators that are already available in the literature. The second-order reduced bias estimators that are considered are also studied for finite samples, through Monte Carlo techniques, as well as applied to real data in the field of finance.