## Publications

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A
Caeiro, Frederico, and Dora Susana Raposo Prata Gomes. "Adaptive estimation of a tail shape second order parameter." International Conference of Computational Methods in Sciences and Engineering 2015 (ICCMSE 2015). AIP Conference Proceedings. American Institute of Physics Inc., 2015. Abstract

In Statistics of Extremes, the tail shape second order parameter is a relevant parameter whenever we want to improve the estimation of first order parameters. We shall consider two semi-parametric estimators of the shape second order parameter, parameterized with a tuning parameter. We provide a Monte Carlo comparative simulation study of several algorithms for the choice of such tuning parameter and for an adaptive estimation of the shape second order parameter.In Statistics of Extremes, the tail shape second order parameter is a relevant parameter whenever we want to improve the estimation of first order parameters. We shall consider two semi-parametric estimators of the shape second order parameter, parameterized with a tuning parameter. We provide a Monte Carlo comparative simulation study of several algorithms for the choice of such tuning parameter and for an adaptive estimation of the shape second order parameter.

Lita da Silva, J., F. Caeiro, I. Natário, and C. A. Braumann Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Berlin Heidelberg: Springer, 2013.productflyer_978-3-642-34903-4.pdf
Caeiro, F., M. I. Gomes, and D. Pestana Alguns resultados adicionais sobre a variância de um estimador de viés reduzido do índice de cauda.. Eds. Correia Ferreira Dias Braumann E. F. S. e Oliveira, I. Actas do XVI Congresso Anual da Sociedade Portuguesa de Estatística - "Arte de Explicar o Acaso"., 2009. Abstract2009spe_art016.pdf

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F., Caeiro, Gomes, and M.I. "Asymptotic Comparison at Optimal Levels of Minimum-Variance Reduced-Bias Tail-Index Estimators." Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Studies in Theoretical and Applied Statistics. Springer Berlin Heidelberg, 2013. 83-91. Abstract
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Caeiro, Frederico, and M.Ivette Gomes. "An asymptotically unbiased moment estimator of a negative extreme value index." Discuss. Math., Probab. Stat.. 30 (2010): 5-19. Abstract

{Summary: We consider a new class of consistent semi-parametric estimators of a negative extreme value index, based on the set of the $k$ largest observations. This class of estimators depends on a control or tuning parameter, which enables us to have access to an estimator with a null second-order component of asymptotic bias, and with a rather interesting mean squared error, as a function of $k$. We study the consistency and asymptotic normality of the proposed estimators. Their finite sample behaviour is obtained through Monte Carlo simulation.}

B
Caeiro, Frederico, and Ivette M. Gomes. "Bias reduction in the estimation of a shape second-order parameter of a heavy-tailed model." Journal Of Statistical Computation And SimulationJournal Of Statistical Computation And Simulation. 85.17 (2015): 3405-3419. AbstractWebsite

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.}

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.} Gomes, Ivette M., Frederico Caeiro, Lígia Henriques-Rodrigues, and B. g Manjunath. "Bootstrap Methods in Statistics of Extremes." Extreme Events in Finance. John Wiley & Sons, Inc., 2016. 117-138. Abstract n/a C Caeiro, F., and M. I. Gomes Caudas pesadas: t de Student e variante assimétrica versus metodologia semi-paramétrica.. Actas do XV Congresso Anual da Sociedade Portuguesa de Estatística - “Da Teoria à Prática”. Lisboa, 2008.art053.pdf 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.} F., Caeiro, and Gomes M.I. "A Class of Semi-parametric Probability Weighted Moment Estimators." Recent Developments in Modeling and Applications in Statistics. Studies in Theoretical and Applied Statistics. Springer Berlin Heidelberg, 2013. 139-147. Abstract n/a D Mateus, Ayana Maria Xavier Furtado, and Frederico Almeida Gião Gonçalves Caeiro. "The difference-sign randomness test." NTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING 2015. Vol. 1702. AIP Conference Proceedings, 1702. American Institute of Physics Inc., 2015. Abstract In this paper we review the properties of the difference-sign randomness test. First we analyse the exact andasymptotic distribution of the test statistic and provide a table with values for the exact distribution function, for samples ofsize n ≤ 32. Then, we also present several moments of the statistic test, under the null hypothesis of randomness and underthe hypothesis of the existence of a linear trend. Finally, we present an illustration of the test difference-sign to a real data set.In this paper we review the properties of the difference-sign randomness test. First we analyse the exact andasymptotic distribution of the test statistic and provide a table with values for the exact distribution function, for samples ofsize n ≤ 32. Then, we also present several moments of the statistic test, under the null hypothesis of randomness and underthe hypothesis of the existence of a linear trend. Finally, we present an illustration of the test difference-sign to a real data set. Caeiro, Frederico, M.Ivette Gomes, and Dinis Pestana. "Direct reduction of bias of the classical Hill estimator." REVSTAT. 3 (2005): 113-136. Abstract {Summary: We are interested in an adequate estimation of the dominant component of the bias of ıt B. M. Hill}\,'s estimator [Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] of a positive tail index$\gamma$, in order to remove it from the classical Hill estimator in different asymptotically equivalent ways. If the second order parameters in the bias are computed at an adequate level$k_1$of a larger order than that of the level$k$at which the Hill estimator is computed, there may be no change in the asymptotic variances of these reduced bias tail index estimators, which are kept equal to the asymptotic variance of the Hill estimator, i.e., equal to$\gamma^2$. The asymptotic distributional properties of the proposed estimators of$\gamma\$ are derived and the estimators are compared not only asymptotically, but also for finite samples through Monte Carlo techniques.}

E
Gomes, M. I., and F. Caeiro Eficiency of partially reduced-bias mean-of-order-p versus minimum-variance reduced-bias extreme value index estimation. COMPSTAT 2014: 21th International Conference on Computational Statistics. Geneve, 2014. Abstractgomes_caeiro_compstat2014_reprint.pdf

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Caeiro, F., and M. I. Gomes Estimação de quantis elevados em estatística de extremos. Actas do XIII Congresso Anual da Sociedade Portuguesa de Estatística - "Ciência Estatística"., 2006.2006spe217-228.pdf
Caeiro, Frederico Almeida Gião Gonçalves, Ayana Maria Xavier Furtado Mateus, and Luís Pedro Carneiro Ramos. "Extreme value analysis of the sea levels in Venice." PROCEEDINGS OF THE INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2014. AIP Conference Proceedings. American Institute of Physics Inc., 2015. Abstract

The number of floods in the city of Venice has increased substantially in the last decades and can be explained bythe sea level rise and land subsidence. Using Statistics of Extremes we shall model the extreme behaviour of the sea level inVenice and quantify risk through the estimation of important parameters such as return periods of high levels.The number of floods in the city of Venice has increased substantially in the last decades and can be explained bythe sea level rise and land subsidence. Using Statistics of Extremes we shall model the extreme behaviour of the sea level inVenice and quantify risk through the estimation of important parameters such as return periods of high levels.

F
Caeiro, Frederico, Ana P. Martins, and Inês J. Sequeira. "Finite sample behaviour of classical and quantile regression estimators for the Pareto distribution." Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014, ICNAAM 2014. Vol. 1648. American Institute of Physics Inc., 2015. Abstract

The Pareto distribution is a well known and important model in Statistics. It has been used to study large incomes, city population size, size of losses, stock price fluctuations, number of citations received by papers and other similar phenomena. In this work we compare the finite sample performance of several estimation methods, namely the Moment, Maximum Likelihood and Quantile Regression methods. The comparison will be made through a Monte-Carlo simulation study.The Pareto distribution is a well known and important model in Statistics. It has been used to study large incomes, city population size, size of losses, stock price fluctuations, number of citations received by papers and other similar phenomena. In this work we compare the finite sample performance of several estimation methods, namely the Moment, Maximum Likelihood and Quantile Regression methods. The comparison will be made through a Monte-Carlo simulation study.

L
Caeiro, F., M. I. Gomes, and L. Henriques-Rodrigues A location invariant probability weighted moment EVI-estimator. Notas e Comunicações do CEAUL 30/2013, 2013.2013_30_port-ppwm-final.pdf
Caeiro, Frederico Almeida Gião Gonçalves, Ivette M. Gomes, and Lígia Henriques-Rodrigues. "A location-invariant probability weighted moment estimation of the Extreme Value Index." International Journal of Computer Mathematics. 93.4 (2016): 676-695. AbstractWebsite

The peaks over random threshold (PORT) methodology and the Pareto probability weighted moments (PPWM) of the largest observations are used to build a class of location-invariant estimators of the Extreme Value Index (EVI), the primary parameter in statistics of extremes. The asymptotic behaviour of such a class of EVI-estimators, the so-called PORT-PPWM EVI-estimators, is derived, and an alternative class of location-invariant EVI-estimators, the generalized Pareto probability weighted moments (GPPWM) EVI-estimators is considered as an alternative. These two classes of estimators, the PORT-PPWM and the GPPWM, jointly with the classical Hill EVI-estimator and a recent class of minimum-variance reduced-bias estimators are compared for finite samples, through a large-scale Monte-Carlo simulation study. An adaptive choice of the tuning parameters under play is put forward and applied to simulated and real data sets.The peaks over random threshold (PORT) methodology and the Pareto probability weighted moments (PPWM) of the largest observations are used to build a class of location-invariant estimators of the Extreme Value Index (EVI), the primary parameter in statistics of extremes. The asymptotic behaviour of such a class of EVI-estimators, the so-called PORT-PPWM EVI-estimators, is derived, and an alternative class of location-invariant EVI-estimators, the generalized Pareto probability weighted moments (GPPWM) EVI-estimators is considered as an alternative. These two classes of estimators, the PORT-PPWM and the GPPWM, jointly with the classical Hill EVI-estimator and a recent class of minimum-variance reduced-bias estimators are compared for finite samples, through a large-scale Monte-Carlo simulation study. An adaptive choice of the tuning parameters under play is put forward and applied to simulated and real data sets.

Caeiro, Frederico, and Dora Susana Raposo Prata Gomes. "A log probability weighted moment estimator of extreme quantiles." Theory and Practice of Risk Assessment - ICRA5 2013. Vol. 136. Springer New York LLC, 2015. 293-303. Abstract

In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.

M
Caeiro, Frederico, Ivette M. Gomes, Jan Beirlant, and Tertius de Wet. "Mean-of-order p reduced-bias extreme value index estimation under a third-order framework." ExtremesExtremes. 19.4 (2016): 561-589. AbstractWebsite

Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-of-order-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.Reduced-bias versions of a very simple generalization of the ‘classical’ Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-of-order-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.