<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Caeiro, Frederico</style></author><author><style face="normal" font="default" size="100%">Gomes, M. Ivette</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Bias reduction in the estimation of parameters of rare events.</style></title><secondary-title><style face="normal" font="default" size="100%">Theory of Stochastic Processes</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">estimators</style></keyword><keyword><style  face="normal" font="default" size="100%">heavy tails}</style></keyword><keyword><style  face="normal" font="default" size="100%">rare events</style></keyword><keyword><style  face="normal" font="default" size="100%">{bias reduction</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2002</style></year></dates><urls><related-urls><url><style face="normal" font="default" size="100%">https://docentes.fct.unl.pt/sites/default/files/fac/files/2002tsp_caeiro_gomes.pdf</style></url></related-urls></urls><volume><style face="normal" font="default" size="100%">8</style></volume><pages><style face="normal" font="default" size="100%">67-76</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;{Consider the distribution function $EV_{\gamma}(x)=\exp(-(1+\gamma x)^{- 1/\gamma}),\ \gamma&amp;gt;0,\ 1+\gamma x&amp;gt;0$, to which $\max(X_{1},łdots, X_{n})$ is attracted after suitable linear normalization. The authors consider the underlying model $F$ in the max-domain of attraction of $EV_{\gamma}$, where $ X_{i:n},\ 1łeq Iłeq n$, denotes the i-th ascending order statistic associated to the random sample $(X_{1},łdots, X_{n})$ from the unknown distribution function $F$. This article is devoted to studying semi-parametric estimators of $\gamma$ in the form $$\gamma_{n}^{(þeta,\alpha)}(k)=(\Gamma(\alpha)/M_{n}^{(\alpha- 1)}(k))łeft(M_{n}^{(þeta\alpha)}(k)/\Gamma(þeta\alpha+1)\right) ^{1/þeta},\quad \alpha\geq 1,\quad þeta&amp;gt;0,$$ parametrized by the parameters $\alpha$ and $þeta$, which may be controlled, where $M_{n}^{(0)}=1$ and $ M_{n}^{(\alpha)}(k)=k^{-1}\sum_{i=1}^{k}(łn X_{n-i+1:n}-łn X_{n-k:n})^{\alpha}$, $\alpha&amp;gt;0$, is a consistent estimator of $\Gamma(\alpha+1)\gamma^{\alpha}$, as $k\toınfty$, and $k=o(n)$, as $n\toınfty$.\par The authors derive the asymptotic distributional properties of the considered class of estimators and obtain that for $þeta&amp;gt;1$ it is always possible to find a control parameter $\alpha$ which makes the dominant component of the asymptotic bias of the proposed estimator null and depends on the second order parameter $\rho$. An investigation of the $\rho$-estimator is presented.}&lt;/p&gt;
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