In Statistics of Extremes, the estimation of the extreme value index is an essential requirement for further tail inference. In this work, we deal with the estimation of a strictly positive extreme value index from a model with a Pareto-type right tail. Under this framework, we propose a new class of weighted Hill estimators, parameterized with a tuning parameter a. We derive their non-degenerate asymptotic behavior and analyze the influence of the tuning parameter in such result. Their finite sample performance is analyzed through a Monte Carlo simulation study. A comparison with other important extreme value index estimators from the literature is also provided.
Abstract. Many insurance premium principles are defined and various estimation procedures introduced in the literature. In this paper, we focus on the estimation of the excessof-loss reinsurance premium when the risks are randomly right-censored. The asymptotic normality of the proposed estimator is established under suitable conditions and its performance evaluated through sets of simulated data.Résumé. Plusieurs principes de calcul de prime d'assurance sont définis et diverses procédures d'estimation sont introduites dans la littérature. Dans cet article, nous nous concentrons sur l'estimation de la prime de réassurance en excédent de sinistres lorsque les risques sont aléatoirement censurésà droite. La normalité asymptotique de l'estimateur proposé estétablie sous des conditions adéquates et sa performanceévaluéeà travers des ensembles de données simulées.
Many insurance premium principles are defined and various estimation procedures introduced in the literature. In this paper, we focus on the estimation of the excessof-loss reinsurance premium when the risks are randomly right-censored. The asymptotic normality of the proposed estimator is established under suitable conditions and its performance evaluated through sets of simulated data.
A tail empirical process for heavy-tailed and right-censored data is introduced and its Gaussian approximation is established. In this context, a (weighted) new Hill-type estimator for positive extreme value index is proposed and its consistency and asymptotic normality are proved by means of the aforementioned process in the framework of second-order conditions of regular variation. In a comparative simulation study, the newly defined estimator is seen to perform better than the already existing ones in terms of both bias and mean squared error. As a real data example, we apply our estimation procedure to evaluate the tail index of the survival time of Australian male Aids patients. It is noteworthy that our approach may also serve to develop other statistics related to the distribution tail such as second-order parameter and reduced-bias tail index estimators. Furthermore, the proposed tail empirical process provides a goodness-of-fit test for Pareto-like models under censorship.
The central limit theorem introduced by Stute [The central limit theorem under random censorship. Ann. Statist. 1995; 23: 422-439] does not hold for some class of heavy-tailed distributions. In this paper, we make use of the extreme value theory to propose an alternative estimating approach of the mean ensuring the asymptotic normality property. A simulation study is carried out to evaluate the performance of this estimation procedure.
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