## Publications

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Cabral, Ivanilda, Frederico Caeiro, and Ivette M. Gomes. "Redução do viés do estimador de Hill: uma nova abordagem." Estatística: Progressos e Aplicações. 2016. 73-84. Abstract
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Cabral, Ivanilda, Frederico Caeiro, and Ivette M. Gomes. "Reduced bias Hill estimators." International Conference of Computational Methods in Sciences and Engineering 2016, ICCMSE 2016. Vol. 1790. American Institute of Physics Inc., 2016. Abstract

For heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.For heavy tails, classical extreme value index estimators, like the Hill estimator, are usually asymptotically biased. Consequently those estimators are quite sensitive to the number of top order statistics used in the estimation. The recent minimum-variance reduced-bias extreme value index estimators enable us to remove the dominant component of asymptotic bias and keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator. In this paper a new minimum-variance reduced-bias extreme value index estimator is introduced, and its non degenerate asymptotic behaviour is studied. A comparison with another important minimum-variance reduced-bias extreme value index estimator is also provided.

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.}

Caeiro, F., and M. I.: Gomes. "A semi-parametric estimator of a shape second order parameter." New Advances in Statistical Modeling and Applications. Eds. Pacheco, A., Santos, R., M. Rosário Oliveira, and C. D. Paulino. Studies in Theoretical and Applied Statistics. Springer, 2014. 137-144. Abstract

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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, and Ivette M. Gomes. "Threshold Selection in Extreme Value Analysis." Extreme Value Modeling and Risk Analysis. Chapman and Hall/CRC 2007, 2016. 69-86. Abstract

The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the extreme value index (EVI). Whenever we are interested in large values, such estimation is usually performed on the basis of the largest k + 1 order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive EVI-estimation. Such a choice can be either heuristic or based on sample paths stability or on the minimization of a mean squared error estimateas a function of k. Some of these procedures will be reviewed. Despite of thefact that the methods provided can be applied, with adequate modifications, to any real EVI and not only to the adaptive EVI-estimation but also to the adaptive estimation of other relevant right-tail parameters, we shall illustrate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI.The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the extreme value index (EVI). Whenever we are interested in large values, such estimation is usually performed on the basis of the largest k + 1 order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive EVI-estimation. Such a choice can be either heuristic or based on sample paths stability or on the minimization of a mean squared error estimateas a function of k. Some of these procedures will be reviewed. Despite of thefact that the methods provided can be applied, with adequate modifications, to any real EVI and not only to the adaptive EVI-estimation but also to the adaptive estimation of other relevant right-tail parameters, we shall illustrate the methods essentially for the EVI and for heavy tails, i.e., for a positive EVI.

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.

Caeiro, F., and M. I. Gomes Second-order Reduced-bias Tail Index and High Quantile Estimation. 56th SESSION OF THE INTERNATIONAL STATISTICAL INSTITUTE. Lisbon, Portugal, 2007.2007isi_volume_lxii_proceedings_caeiro_gomes.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.

Caeiro, Frederico, and M.Ivette Gomes. "Minimum-variance reduced-bias tail index and high quantile estimation." REVSTAT. 6 (2008): 1-20. Abstract

{Summary: Heavy tailed-models are quite useful in many fields, like insurance, finance, telecommunications, internet traffic, among others, and it is often necessary to estimate a high quantile, i.e., a value that is exceeded with a probability $p$, small. The semiparametric estimation of this parameter relies essentially on the estimation of the tail index, the primary parameter in statistics of extremes. Classical semi-parametric estimators of extreme parameters show usually a severe bias and are known to be very sensitive to the number $k$ of top order statistics used in the estimation. For $k$ small they have a high variance, and for large $k$ a high bias. Recently, new second-order shape'' and scale'' estimators allowed the development of second-order reduced-bias estimators, which are much less sensitive to the choice of $k$. Here we study, under a third order framework, minimum-variance reduced-bias (MVRB) tail index estimators, recently introduced in the literature, and dependent on an adequate estimation of second order parameters. The improvement comes from the asymptotic variance, which is kept equal to the asymptotic variance of the classical Hill estimator [ıt B. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided that we estimate the second order parameters at a level of a larger order than the level used for the estimation of the first order parameter. The use of those MVRB tail index estimators enables us to introduce new classes of reduced-bias high quantile estimators. These new classes are compared among themselves and with previous ones through the use of a small-scale Monte Carlo simulation.}

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.

Caeiro, Frederico, and M.Ivette Gomes. "Semi-parametric second-order reduced-bias high quantile estimation." Test. 18 (2009): 392-413. Abstract

{Summary: In many areas of application, like, for instance, climatology, hydrology, insurance, finance, and statistical quality control, a typical requirement is to estimate a high quantile of probability $1 - p$, a value high enough so that the chance of an exceedance of that value is equal to $p$, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index $\gamma$, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for $\gamma >0$, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles' estimators, with the same asymptotic variance as the (biased) classical'' estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.}

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.

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, 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
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, 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.

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.}

Caeiro, F., and M. I. Gomes. "Probability weighted moments bootstrap estimation: a case study in the field of insurance." Risk and Extreme Values in Insurance and Finance: Book of Abstracts. Lisbon: CEAUL, 2011. 27-30.rev2011_caeiro_gomes.pdf
Caeiro, Frederico, and M.Ivette Gomes. "A new class of estimators of a scale'' second order parameter." Extremes. 9 (2006): 193-211. Abstract

{Let $X_i$ be i.i.d. r.v.s with heavy-tailed CDF $F(x)$ such that $$1-F(x)=(x/C)^{-1/\gamma}((1+(\beta/\rho)(x/C)^{\rho/\gamma} +\beta'(x/C)^{2\rho/\gamma}(1+o(1))),$$ where $\gamma$ is the tail index ($\gamma>0$), and $\rho<0$ and $\beta$ are the second order parameters''. The authors construct an estimator for $\beta$ based on the tail moments'' $$M_n^{(\alpha)}=(k)^{-1}\sum_{i=1}^k [łog X_{n-i+1:n}-łog X_{n-k:n}]^\alpha.$$ Consistency and asymptotic normality of the estimator are demonstrated. The small sample properties of the estimator are investigated via simulations.}

Caeiro, Frederico, M.Ivette Gomes, and Lígia Henriques Rodrigues. "Reduced-bias tail index estimators under a third-order framework." Commun. Stat., Theory Methods. 38 (2009): 1019-1040. Abstract

{Summary: We are interested in the comparison, under a third-order framework, of classes of second-order, reduced-bias tail index estimators, giving particular emphasis to minimum-variance reduced-bias estimators of the tail index $\gamma$. The full asymptotic distributional properties of the proposed classes are derived under a third-order framework and the estimators are compared with other alternatives, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the USA Dollar is also provided.}