Browsing by Author "Alves, M. Isabel Fraga"
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- Peaks over random threshold methodology for tail index and high quantile estimationPublication . Santos, Paulo Araújo; Alves, M. Isabel Fraga; Gomes, M. IvetteIn this paper we present a class of semi-parametric high quantile estimators which enjoy a desirable property in the presence of linear transformations of the data. Such a feature is in accordance with the empirical counterpart of the theoretical linearity of a quantile χp: χp(δX + λ) = δχp(X) + λ, for any real λ and positive δ. This class of estimators is based on the sample of excesses over a random threshold, originating what we denominate PORT (Peaks Over Random Threshold) methodology. We prove consistency and asymptotic normality of two high quantile estimators in this class, associated with the PORT-estimators for the tail index. The exact performance of the new tail index and quantile PORT-estimators is compared with the original semiparametric estimators, through a simulation study.
- PORT Hill and Moment Estimators for Heavy-Tailed ModelsPublication . Gomes, M. Ivette; Alves, M. Isabel Fraga; Santos, Paulo AraújoIn this article, we use the peaks over random threshold (PORT)-methodology, and consider Hill and moment PORT-classes of extreme value index estimators. These classes of estimators are invariant not only to changes in scale, like the classical Hill and moment estimators, but also to changes in location. They are based on the sample of excesses over a random threshold, the order statistic X[np]+1:n, 0 ≤ p < 1, being p a tuning parameter, which makes them highly flexible. Under convenient restrictions on the underlying model, these classes of estimators are consistent and asymptotically normal for adequate values of k, the number of top order statistics used in the semi-parametric estimation of the extreme value index γ. In practice, there may however appear a stability around a value distant from the target γ when the minimum is chosen for the random threshold, and attention is drawn for the danger of transforming the original data through the subtraction of the minimum. A new bias-corrected moment estimator is also introduced. The exact performance of the new extreme value index PORT-estimators is compared, through a large-scale Monte-Carlo simulation study, with the original Hill and moment estimators, the bias-corrected moment estimator, and one of the minimum-variance reduced-bias (MVRB) extreme value index estimators recently introduced in the literature. As an empirical example we estimate the tail index associated to a set of real data from the field of finance.