# Lígia Henriques RodriguesUniversidade de Évora | uevora · Department of Mathematics

Lígia Henriques Rodrigues

Ph.D. In Statistics and Operations Research

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142

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Introduction

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February 2015 - present

March 1997 - December 2014

## Publications

Publications (142)

Statistics of extremes are nowadays faced with many challenges, among which we refer the ones related to topics like risk modelling of big data and robustness of the methodologies that enable to understand the complexity of extreme events in the most diverse fields. Since an important situation in risk management is the risk of a big loss, a great...

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation...

In the field of statistical extreme value theory, a great variety of alternative methodologies are available to deal with the management of risks of extreme events. Indeed, an important situation in risk management is the risk of a big loss that occurs very rarely. The risk is generally expressed either by the value at risk at a level q (VaRq) or b...

The Box-Cox transformations are used to make the data more suitable for statistical analysis. We know from the literature that this transformation of the data can increase the rate of convergence of the tail of the distribution to the generalized extreme value distribution, and as a byproduct, the bias of the estimation procedure is reduced. The re...

Most of the estimators of parameters of rare and large events, among which we distinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory (EVT), are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarit...

The Hill estimator, one of the most popular extreme value index (EVI) estimators under a heavy right-tail framework, i.e. for a positive EVI, here denoted by ξ, is an average of the log-excesses. Consequently, it can be regarded as the logarithm of the geometric mean or mean of order p = 0 of an adequate set of systematic statistics. We can thus mo...

The Weibull tail-coeffcient (WTC) is the parameter in a right-tail function such that H := ln (1-F) is a regularly varying function at infinity with an index of regular variation equal to 1/WTC. In a context of extreme value theory for maxima, it is possible to prove that we have an extreme value index (EVI) equal to 0, but usually a very slow rate...

The role of generalized means in the estimation of the Weibull tail coefficient is put forward.

Extreme value theory (EVT) helps us to potentially control disastrous events of high relevance to society. Floods, fires, hurricanes, and other extreme occurrences have provided impetus for new developments of EVT. And several generalised means (GMs) have recently been used with success in the estimation of a positive extreme value index (EVI). Det...

The reduction of bias of the Hill estimator has been extensively addressed in the literature of extreme value theory. Several techniques have been used to achieve such reduction of bias, either by removing the main component of the bias of the Hill estimator of the extreme value index (EVI) or by constructing new estimators based on generalized mea...

Statistics of extremes, either univariate or multivariate, have been recently faced with many challenges, especially the ones related to topics like risk modelling of big data and robustness of the methodologies that enable to understand the complexity of extreme events in the most diverse areas of applications. In statistical extreme value theory...

The Hill estimator, one of the most popular extreme value index (EVI) estimators under a heavy right-tail framework, i.e. for a positive EVI, is an average of the log-excesses. Consequently, the Hill EVI-estimator can be regarded as the logarithm of the geometric mean, i.e. the mean of order p = 0 of a set of simple statistics related to the log-ex...

Most of the estimators of parameters of rare and large events are related to averages of statistics based on the k upper observations. And those averages can obviously be regarded as the logarithm of the geometric mean, i.e. the power mean of order p=0 of a certain set of adequate associated statistics. Among the aforementioned parameters, we disti...

O estimador de Hill é o estimador mais popular de uḿ ındice de valores extremos (EVI, do inglês 'extreme value index') positivo. Trata-se de uma média aritmética, sendo consequentemente o logaritmo da média geométrica, i.e. da média-de-ordem-0, de estatísticas adequadas, função das estatísticas ordinais de topo associadas a uma amostra aleatória. P...

Statistics of extremes helps us to control potentially disastrous events, of a high relevance for society and a high social impact, like earthquakes. There are usually only a few observations in the tail of the distribution underlying the data. Estimates much below/above the observed minimum/maximum are required, and we thus need to consider reliab...

Extreme value theory (EVT) helps us to potentially control disastrous events. Floods, fires, hurricanes and other extreme occurrences have provided impetus for new developments of EVT. Generalized means (GM) have recently been used with success in the estimation of a positive extreme value index (EVI). Due to the specificity of the Weibull tail coe...

An average of statistics Si can be regarded as the logarithm of the geometric mean, or the power mean of order p = 0, of exp(Si), 1 ≤ i ≤ k. Instead of such a geometric mean, we can more generally consider the power mean of order p (MOp) of those statistics, p ∈ R, and to build a class of MOp-estimators. The Hill estimators, average of the log-exce...

Statistics of extremes (SE) are today faced with many challenges, especially the ones related to topics like risk modeling of big data and robustness of the methodologies that enable to understand the complexity of extreme events in the most diverse areas. Generalized means (GMs) have recently been used with success in the estimation of a positive...

O estimador de Hill é o estimador mais popular de um índice de valores extremos (EVI, do inglês 'extreme value index') positivo, denotado por ξ. Trata-se de uma média aritmética, sendo consequentemente o logaritmo da média geométrica, i.e. da média-de-ordem-0, de estatísticas adequadas, função das estatísticas ordinais de topo associadas a uma amos...

Most of the estimators of parameters of rare and large events, among which we distinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory (EVT), are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarit...

In Extreme Value Analysis, the extreme value index, ξ, is the primary parameter of extreme events. In this work, we consider ξ positive, i.e. we assume that F is heavy tailed. Classical tail parameters estimators, such as the Hill, the Moments or the Weissman estimators, are usually asymptotically biased. Consequently, those estimators are quite se...

On the basis of a sample of either independent, identically distributed or possibly weakly dependent and stationary random variables from an unknown model F with a heavy right-tail function, and for any small level q, the value-at-risk (VaR) at the level q, i.e. the size of the loss that occurs with a probability q, is estimated by new semi-paramet...

Statistics of extremes helps us to control potentially disastrous events, of a high relevance for society and a high social impact, like earthquakes. There are usually only a few observations in the tail of the distribution underlying the data. Estimates much below/above the observed minimum/maximum are required, and we thus need to consider reliab...

In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill es...

In Statistics of Extremes we often have to deal with the estimation of the extreme value index, a key parameter of extreme events. The adequate estimation of this parameter is of crucial importance in the estimation of other parameters of extreme events, such as an extreme quantile, a small exceedance probability or the return period of a high leve...

In this work we propose a new class of consistent semi-parametric tail index estimators for light-tailed models, i.e., models with a negative extreme value index. Light-tailed models are very common in practice, in areas such as environment and hydrology, among others. The extreme value index is the primary parameter in statistics of extremes and i...

The International Swimming Federation has developed a points system that allows comparisons of results between different events. Such system is important for several reasons, since it is used as a criterion to rank swimmers in awards and selection procedures of national teams. The points system is based entirely on the world record of the correspon...

On the basis of partially reduced-bias and reduced-bias power-mean-of-order-p estimators of the extreme value index, we advise the consideration of new estimators of the value-at-risk. After a brief reference to the asymptotic properties of these new VaR-estimators, we proceed to an overall comparison of VaR-estimators, through Monte-Carlo simulati...

We are interested in reduced-bias (RB) estimators of the Value at Risk at a level q (VaR_q), a value, high enough, so that the chance of an exceedance of that value is equal to q, small. We shall deal only with heavy tails, trying to improve the performance of the existent RB VaR-estimators. The VaR, i.e. a high quantile, depends on the extreme val...

Under a convenient third-order framework, the asymptotic distributional behaviour of a class of location invariant reduced-bias tail index estimators is derived. Such a class is based on the PORT methodology, with PORT standing for peaks over random thresholds, and combines a PORT-version of one of the pioneering classes of minimum-variance reduced...

Under a convenient third-order framework, the asymptotic distributional behavior of a class of location invariant reduced-bias tail index estimators is derived. Such a class is based on the PORT methodology, with PORT standing for peaks over random thresholds, and combines a PORT-version of one of the pioneering classes of minimum-variance reduced...

This is a correction to a figure in the article: Gomes, M.I. & Henriques-Rodrigues, L. (2016). Competitive estimation of the extreme value index. Statist. and Probab. Letters 117, 128-135.

In many areas of application, it is a common practice to estimate the value at risk at a level q (VaR_q), a value, high enough, so that the chance of an exceedance of that value is equal to q, small, often smaller than 1/n, with n the size of available sample. For heavy-tailed models, quite common in many areas of application, like biostatistics, f...

New classes of reliable extreme value index (EVI)-estimators based on adequate generalized means (GM) have recently appeared in the literature, and will be introduced. The use of these GM EVI-estimators has enabled the introduction of new and interesting classes of GM value-at-risk (VaR)-estimators. But the GM EVI-estimators are NOT location-invari...

In many areas of application, like environment, finance, insurance, statistical quality control, and on the basis of a transformed sample, which can be considered weakly dependent and stationary from an unknown model F, it is a common practice to estimate different parameters of extreme events. Among them, we refer the valueat-risk (VaR) at a small...

In many areas of application, like environment, finance, insurance, statistical quality control, and on the basis of a transformed sample, which can be considered weakly dependent and stationary from an unknown model F , it is a common practice to estimate different parameters of extreme events. Among them, we refer the value-at-risk (VaR) at a sma...

Under a convenient third-order framework, the asymptotic distributional behavior of a class of location invariant reduced-bias tail index estimators is derived. Such a class is based on the PORT methodology, with PORT standing for peaks over random thresholds, and combines a PORT-version of one of the pioneering classes of minimum-variance reduced...

The value-at-risk (VaR) at a small level q, 0 < q < 1, is the size of the loss that occurs with a probability q. Semi-parametric partially reduced-bias (PRB) VaR-estimation procedures based on the mean-of-order-p of a set of k quotients of upper order statistics, with p any real number, are put forward. After the study of their asymptotic behaviour...

In finance, insurance and statistical quality control, among many other areas of application , a typical requirement is to estimate the value-at-risk (VaR) at a small level q, i.e. a high quantile of probability 1 − q, a value, high enough, so that the chance of an exceedance of that value is equal to q, small. The semi-parametric estimation of hig...

In many areas of application, like environment, finance, insurance and statistical quality control, and on the basis of a sample of either independent, identically distributed or possibly weakly dependent and stationary random variables from an unknown model F, it is a common practice to estimate the value-at-risk (VaR) at a small level q, i.e. a...

In many areas of application, it is a common practice to estimate the Value at Risk at a level q (VaR q), a value, high enough, so that the chance of an exceedance of that value is equal to q, small, often ≤ 1/n, with n the sample size. We here deal only with heavy tails, trying to improve the performance of the existent VaR-estimators.

The value-at-risk (VaR) at a small level q, 0 < q << 1, is the size of the loss that occurs with a probability q. In the lines of Weissman's semi-parametric VaR-estimator, and based upon the extreme value index estimators in recent articles by the authors, semi-parametric partially reduced-bias (PRB) VaR-estimation procedures based on the mean-of-o...

In this chapter we provide an overview of the bootstrap methodology together with its possible use in the reliable estimation of any parameter of extreme events. For an asymptotically consistent choice of the threshold to use in the estimation of the extreme value index (EVI),we suggest and discuss the so-called double-bootstrap algorithm, where in...

In this chapter we provide an overview of the bootstrap methodology together with its possible use in the reliable estimation of any parameter of extreme events. For an asymptotically consistent choice of the threshold to use in the estimation of the extreme value index (EVI), we suggest and discuss the so-called double-bootstrap algorithm, where i...

In finance, insurance and statistical quality control, among many other areas of application , a typical requirement is the estimation of the value-at-risk (VaR) at a small level q, i.e. a high quantile of probability 1 − q, a value, high enough, so that the chance of an exceedance of that value is equal to q, small. The semi-parametric estimation...

A simple generalisation of the classical Hill estimator of a positive extreme value index (EVI) has been recently introduced in the literature. Indeed, the Hill estimator can be regarded as the logarithm of the mean of order p = 0 of a certain set of statistics. Instead of such a geometric mean, we can more generally consider the mean of order p (M...

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...

The mean-of-order-p (MO p) extreme value index (EVI) estimators are based on Hölder's mean of an adequate set of statistics, and generalize the classical Hill EVI-estimators, associated with p = 0. Such a class of estimators, dependent on the tuning parameter p ∈ R, has revealed to be highly flexible, but it is not invariant for changes in location...

The mean-of-order- ( ) extreme value index (EVI) estimators are based on Hölder’s mean of an adequate set of statistics, and generalize the classical Hill EVI-estimators, associated with . Such a class of estimators, dependent on the tuning parameter , has revealed to be highly flexible, but it is not invariant for changes in location. To make the...

Given a sample of size n of either independent, identically distributed or possibly stationary weakly dependent random variables from a cumulative distribution function (CDF) F , let us assume that F is in the domain of attraction for maxima of an extreme value (EV) CDF with a positive extreme value index (EVI). For this type of Pareto right-tailed...

For heavy right tails and under a semi-parametric framework, we introduce a class of location invariant estimators of a scale second-order parameter and study its asymptotic non-degenerate behaviour. This class is based on the PORT methodology, with PORT standing for peaks over random thresholds. The consistency and asymptotic normality of the new...

A metodologia PORT (do Inglês, ‘peaks over a random threshold’), dependente de q, em [0,1), é aplicada aos estimadores do índice de cauda baseados numa média generalizada de ordem p. Avançamos com algoritmos para a escolha do vector de parâmetros de controlo, (k, p, q), em que k é o número de estatísticas ordinais em jogo, e aplicamos a metodologia...

Resampling computer intensive methodologies, like the jackknife and the bootstrap are important tools for a reliable semi-parametric estimation of parameters of extreme or even rare events. Among these parameters we mention the extreme value index, ξ, the primary parameter in statistics of extremes. Most of the semi-parametric estimators of this pa...

Tendo em vista a aplicação a dados de melhores marcas em modalidades de atletismo, daremos atenção à estimação do índice de cauda, bem como à estimação do limite superior do suporte, se finito, o “recorde mundial” possível face às condições atuais.

A simple generalisation of the classical Hill estimator of a positive extreme value index (EVI) has been recently introduced in the literature. Indeed, the Hill estimator can be regarded as the logarithm of the mean of order p = 0 of a certain set of statistics. Instead of such a geometric mean, we can more generally consider the mean of order p (M...

In this paper we study, under a semi-parametric framework and for heavy right tails, a class of location invariant estimators of a shape second-order parameter, ruling the rate of convergence of the normalised sequence of maximum values to a non-degenerate limit. This class is based on the PORT methodology, with PORT standing for peaks over random...

For heavy right tails and under a semi-parametric framework, we introduce a class of location invariant estimators of a scale second-order parameter and study its asymptotic non-degenerate behaviour. This class is based on the PORT methodology, with PORT standing for peaks over random thresholds. The consistency and asymptotic normality of the new...

In this chapter we provide an overview of the bootstrap methodology together with its possible use in the reliable estimation of any parameter of extreme or even rare events. For an asymptotically consistent choice of the thresholds to use in the estimation of the extreme value index (EVI), ξ, we suggest and discuss a double-bootstrap algorithm for...

A new class of location invariant estimators of a positive extreme value index (EVI) is introduced. On the basis of second-order best linear unbiased estimators of the EVI, a class of PORT best linear EVI-estimators is considered, with PORT standing for peaks over random thresholds. A heuristic procedure for the adaptive choice of the tuning parame...

Resampling computer intensive methodologies, like the jack-knife and the bootstrap are important tools for a reliable semi-parametric estimation of parameters of extreme or even rare events. Among these parameters we mention the extreme value index, γ, the primary parameter in statistics of extremes. Most of the semi-parametric estimators of this p...

The peaks over random threshold (PORT) methodology and the Pareto probabil-
ity weighted moments (PPWM) of 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 P...

Under a semi-parametric framework and for heavy right tails, we introduce a class of location invariant estimators of an adequate shape second-order parameter, also ruling the rate of convergence of a normalized sequence of maximum values to a nondegenerate limit. This class is based on the PORT methodology, with PORT standing for peaks over random...

In this chapter, we consider an application to environmental data of a bootstrap algorithm for the adaptive estimation of the extreme value index (EVI), the primary parameter in Statistics of Extremes. The EVI estimation is performed through the recent Peaks Over Random Threshold Minimum-Variance Reduced-Bias (PORT-MVRB) estimators, which apart fro...

We are interested in reduced-bias versions of a simple generalisation of the classical Hill estimator of a positive extreme value index (EVI), the primary parameter of extreme events. This class is based on the mean of order p (MOP) of adequate statistics. The asymptotic behaviour of the class of MOP EVI-estimators is reviewed, and associated reduc...

In many areas of application, like Insurance and Finance, a typical requirement, under a semi-parametric framework, is the estimation of the Value at Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, and for heavy rigt-tails, the classical VaR estimators are the Weissman-Hill estimators, based...

The main objective of statistics of univariate extremes (SUE) lies in the estimation of quantities related to extreme events. The main parameter in SUE is the extreme value index (EVI), the shape parameter 2 R, in the general extreme value (EV) distribution function. We shall consider heavy tails. Under such a setup , the classical EVI-estimators a...

For heavy right tails and under a semi-parametric framework, we study the asymptotic non-degenerate behaviour of classes of location invariant estimators of the shape and scale second-order parameters. These classes are based on the PORT methodology, with PORT standing for peaks over random thresholds. Asymptotic normality of such estimators is ach...

In this chapter, we consider a recent class of generalized negative moment estimators of a negative extreme value index, the primary parameter in statistics of extremes. Apart from the usual integer parameter k, related to the number of top order statistics involved in the estimation, these estimators depend on an extra real parameter θ, which make...

In this article, we deal with an empirical comparison of two data-driven heuristic procedures of estimation of a positive extreme value index (EVI), working thus with heavy right tails. The semi-parametric EVI estimators under consideration, the so-called peaks over random threshold (PORT)–minimum-variance reduced-bias (MVRB) EVI-estimators, are lo...

For heavy right tails and under a semi-parametric framework, we introduce a class of location invariant estimators of a scale second-order parameter and study its asymptotic degenerate behaviour. This class is based on the PORT methodology, with PORT standing for peaks over random thresholds. Consistency of the new class of estimators is achieved u...

An adequate estimation of the second-order parameters in the class of peaks over random threshold (PORT) minimum variance reduced bias (MVRB) extreme value index (EVI) estimators, which are both location and scale invariant, depends on heuristic choices of some tuning parameters. We explore a new data-driven heuristic procedure in estimating the sc...

Neste trabalho apresentamos uma nova classe de estimadores, semi-paramétricos e invariantes para mudanças de localização, de um parâmetro de escala de segunda ordem. A derivação da nova classe de estimadores baseia-se na aplicação da metodologia PORT, do inglês “peaks over random thresholds ” a uma classe de estimadores clássicos de um parâmetro de...

Through the use of Pareto probability weighted moments (PPWM) based on the k + 1 largest observations, we first refer the PPWM semi-parametric estimators of-the extreme value index (EVI), the primary parameter of extreme events. But these EVI-estimators are NOT location-invariant, contrarily to the PORT-PPWM estimators, which depend on an extra tun...

Making use of the peaks over random threshold (PORT) methodology and the Pareto probability weighted moments (PPWM) of the largest observations, and moreover dealing with the extreme value index (EVI), the primary parameter in statistics of extremes, new classes of location-invariant EVI-estimators are built. These estimators, the so-called PORT-PP...

After an introduction to the PORT-estimation of the shape second-order parameter, denoted by ρ, main asymptotic results of suggested estimators are discussed. The choice of the tuning parameter under play for the PORT-ρ-estimators is provided, though an empirical criterion and an optimal method. An illustration is further provided.

Dealing with the mean of order p ≥ 0 (MOp) of adequate statistics, we are interested in reduced-bias versions of a class of MOp extreme value index (EVI)-estimators, a very simple generalisation of the classical Hill estimator of a positive EVI, the primary parameter of extreme events. The asymptotic behaviour of the class of MOp EVI-estimators is...

In this paper we introduce, under a semi-parametric framework and for heavy right tails,
a class of location invariant estimators of a shape second-order parameter, also ruling the
rate of convergence of the normalised sequence of maximum values to a non-degenerate
limit. This class is based on the PORT methodology, with PORT standing for peaks ove...

In this paper, we deal with the estimation, under a semi-parametric framework, of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, the classical VaR estimators are the Weissman–Hill estimators, based on any intermediate number k of top-order statistics. But these VaR estimators do...

In this article, we deal with an empirical comparison of two data-driven heuristic procedures of estimation of a positive extreme value index (EVI), working thus with heavy right tails. The semi-parametric EVI-estimators under consideration, the so-called peaks over random threshold (PORT)–minimum-variance reduced-bias (MVRB) EVI-estimators, are lo...

Neste artigo abordamos um método de estimação, semi-paramétrico e invariante para
mudanças de localização e escala, de um índice de valores extremos (EVI, do inglês, extreme value
index) positivo. Tomando como base estimadores lineares centrados (BLUE, do Inglês best linear
unbiased estimators) do EVI, consideramos agora estimadores PORT-BLUE do EV...