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2016
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, Filipe J. Marques, Ayana Mateus, and Serra Atal. "A note on the Jackson exponentiality test." International Conference of Computational Methods in Sciences and Engineering 2016, ICCMSE 2016. Vol. 1790. American Institute of Physics Inc., 2016. Abstract

In this paper we revisit the Jackson exponentiality test. We study and provide functions in R language to compute theoretical moments, the distribution function and quantiles of the statistic test. Approximations to the exact distribution function and quantiles are also provided and their precision discussed. In addition, we provide an application of the Jackson test to real data.In this paper we revisit the Jackson exponentiality test. We study and provide functions in R language to compute theoretical moments, the distribution function and quantiles of the statistic test. Approximations to the exact distribution function and quantiles are also provided and their precision discussed. In addition, we provide an application of the Jackson test to real data.

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.

Mateus, Ayana, Frederico Caeiro, Dora Prata Gomes, and Inês J. Sequeira. "Statistical analysis of extreme river flows." International Conference of Computational Methods in Sciences and Engineering 2016, ICCMSE 2016. Vol. 1790. AIP Conference Proceedings, 1790. American Institute of Physics Inc., 2016. Abstract

Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.

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.

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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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
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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.

2015
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, 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. "Preface of the "2nd Symposium on Computational Statistical Methods"." AIP Conference ProceedingsAIP Conference Proceedings. 1702 (2015). AbstractWebsite
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Caeiro, Frederico, and Ivette M. Gomes. "Revisiting the maximum likelihood estimation of a positive extreme value index." Journal Of Statistical Theory And PracticeJournal Of Statistical Theory And Practice. 9.1 (2015): 200-218. AbstractWebsite

In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.In this article, we revisit Feuerverger and Halls maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.

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

In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.

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 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 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.

Gomes, Ivette M., Fátima M. Brilhante, Frederico Caeiro, and Dinis Pestana. "A new partially reduced-bias mean-of-order p class of extreme value index estimators." Computational Statistics & Data AnalysisComputational Statistics & Data Analysis. 82 (2015): 223-227. AbstractWebsite

A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.A class of partially reduced-bias estimators of a positive extreme value index (EVI), related to a mean-of-order-p class of EVI-estimators, is introduced and studied both asymptotically and for finite samples through a Monte-Carlo simulation study. A comparison between this class and a representative class of minimum-variance reduced-bias (MVRB) EVI-estimators is further considered. The MVRB EVI-estimators are related to a direct removal of the dominant component of the bias of a classical estimator of a positive EVI, the Hill estimator, attaining as well minimal asymptotic variance. Heuristic choices for the tuning parameters p and k, the number of top order statistics used in the estimation, are put forward, and applied to simulated and real data.

2014
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. "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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F., Caeiro, Gomes M.I., and Vandewalle B. "Semi-Parametric Probability-Weighted Moments Estimation Revisited." (2014). Abstract

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Caeiro, Frederico Almeida Gião Gonçalves, and Ayana Maria Xavier Furtado Mateus. "An R implementation of several randomness tests." AIP Conference Proceedings. 2014. 531-534. Abstract

In many statistic methods, including distribution-free methods, we assume to work with random samples. In this note, we present randtests: an R package implementation of several nonparametric randomness tests. After a brief description of the tests included in the package, we present an application to real data sets in the field of Agricultural.In many statistic methods, including distribution-free methods, we assume to work with random samples. In this note, we present randtests: an R package implementation of several nonparametric randomness tests. After a brief description of the tests included in the package, we present an application to real data sets in the field of Agricultural.

Caeiro, F., and M. I. Gomes On the bootstrap methodology for the estimation of the tail sample fraction. COMPSTAT 2014: 21th International Conference on Computational Statistics. Geneve, 2014.caeiro_gomes_compstat2014_reprint.pdf
2013
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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