Frederico Caeiro

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{Summary: We are interested in the comparison, under a third-order framework, of classes of second-order, reduced-bias tail index estimators, giving particular emphasis to minimum-variance reduced-bias estimators of the tail index $\gamma$. The full asymptotic distributional properties of the proposed classes are derived under a third-order framework and the estimators are compared with other alternatives, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the USA Dollar is also provided.}

In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.

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{Summary: In many areas of application, like, for instance, climatology, hydrology, insurance, finance, and statistical quality control, a typical requirement is to estimate a high quantile of probability $1 - p$, a value high enough so that the chance of an exceedance of that value is equal to $p$, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index $\gamma $, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for $\gamma >0$, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles' estimators, with the same asymptotic variance as the (biased) ``classical'' estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.}

{Summary: For heavy-tailed models, and working with the sample of the $k$ largest observations, we present probability weighted moments (PWM) estimators for the first order tail parameters. Under regular variation conditions on the right-tail of the underlying distribution function $F$ we prove the consistency and asymptotic normality of these estimators. Their performance, for finite sample sizes, is illustrated through a small-scale Monte Carlo simulation.}

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