## Publications

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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.

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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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.}

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
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.

Caeiro, Frederico, and M.Ivette Gomes. "Semi-parametric tail inference through probability-weighted moments." J. Stat. Plann. Inference. 141 (2011): 937-950. Abstract

{Summary: For heavy-tailed models, and working with the sample of the $k$ largest observations, we present probability weighted moments (PWM) estimators for the first order tail parameters. Under regular variation conditions on the right-tail of the underlying distribution function $F$ we prove the consistency and asymptotic normality of these estimators. Their performance, for finite sample sizes, is illustrated through a small-scale Monte Carlo simulation.}

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. "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. "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, F., and M. I. Gomes Redução de viés na estimação semi-paramétrica de um parâmetro de escala. Actas do XIII Congresso da SPE - "Ciência Estatística"., 2006.2006spe_127-148.pdf
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, 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.}

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
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 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, 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.