Bias reduction of a tail index estimator through an external estimation of the second-order parameter |
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Authors: | M. Ivette Gomes Frederico Caeiro Fernanda Figueiredo |
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Affiliation: | 1. Faculdade de Ciências (DEIO) and CEAUL , Universidade de Lisboa, Edificio c6, Piso 4, Cidade Universitária, Campo Grande , 1749-016, Lisboa, Portugal ivette.gomes@fc.ul.pt;3. Departamento de Matemática , Universidade Nova de Lisboa , FCT, Portugal;4. Faculdade de Economia and CEAUL , Universidade do Porto , Portugal |
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Abstract: | In this paper, we first consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterised in a tuning or control parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of the tail index γ with a null dominant component of asymptotic bias, and consequently with a reasonably flat mean squared error pattern, as a function of k, the number of top-order statistics considered. Such a control parameter depends on a second-order parameter ρ, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator, provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro–US$ exchange rates. |
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Keywords: | Statistical theory of extremes Semi-parametric Estimation |
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