## Publications

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2005
Caeiro, F., and M. I. Gomes Uma classe de estimadores do parâmetro de escala de segunda ordem.. Actas do XII Congresso Anual da Sociedade Portuguesa de Estatística. Évora, Portugal, 2005.caeirof-spe2004.pdf
2004
Gomes, M.Ivette, Frederico Caeiro, and Fernanda Figueiredo. "Bias reduction of a tail index estimator through an external estimation of the second-order parameter." Statistics. 38 (2004): 497-510. Abstract

{Summary: We first consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parametrised in a tuning or control parameter $\alpha$. Such a control parameter enables us to have access for any available sample, to an estimator of the tail index $\gamma$ 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.\par Such a control parameter depends on a second-order parameter $\rho$, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] 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.} 2003 Caeiro, F. Redução de viés em estimadores do índice de cauda. Actas do X Congresso Anual da SPE - “Literacia e Estatística”. Porto, Portugal, 2003.spe2002_187-199.pdf 2002 Caeiro, Frederico, and Ivette M. Gomes. "Bias reduction in the estimation of parameters of rare events." Theory of Stochastic Processes. 8.24 (2002): 67-76. Abstract2002tsp_caeiro_gomes.pdf {Consider the distribution function$EV_{\gamma}(x)=\exp(-(1+\gamma x)^{- 1/\gamma}),\ \gamma>0,\ 1+\gamma x>0$, 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 $$\gamma_{n}^{(þeta,\alpha)}(k)=(\Gamma(\alpha)/M_{n}^{(\alpha- 1)}(k))łeft(M_{n}^{(þeta\alpha)}(k)/\Gamma(þeta\alpha+1)\right) ^{1/þeta},\quad \alpha\geq 1,\quad þeta>0,$$ 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>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>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$-estimator is presented.} Caeiro, Frederico, and Ivette M. Gomes. "A class of asymptotically unbiased semi-parametric estimators of the tail index." Test. 11 (2002): 345-364. Abstract {Summary: We consider a class of consistent semi-parametric estimators of a positive tail index$\gamma$, parameterized by a tuning or control parameter$\alpha$. Such a control parameter enables us to have access, for any available sample, to an estimator of$\gamma$with a null dominant component of asymptotic bias, and with a reasonably flat mean squared error pattern, as a function of$k\$, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)], provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator.}