Ligia Henriques-Rodrigues - Academia.edu (original) (raw)
Papers by Ligia Henriques-Rodrigues
Extremes, 2008
In statistics of extremes, inference is often based on the excesses over a high random threshold.... more In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called "maximum likelihood" estimators of the tail index γ . In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar "maximum likelihood" estimator of a positive tail index γ , based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study.
DOAJ (DOAJ: Directory of Open Access Journals), Jun 1, 2016
• A simple generalisation of the classical Hill estimator of a positive extreme value index (EVI)... more • 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 (MOP) of those statistics, with p real, and even an optimal MOP (OMOP) class of EVI-estimators. These estimators are scale invariant but not location invariant. With PORT standing for peaks over random threshold, new classes of PORT-MOP and PORT-OMOP EVI-estimators are now introduced. These classes are dependent on an extra tuning parameter q, 0 ≤ q < 1, and they are both location and scale invariant, a property also played by the EVI. The asymptotic normal behaviour of those PORT classes is derived. These EVI-estimators are further studied for finite samples, through a Monte-Carlo simulation study. An adequate choice of the tuning parameters under play is put forward, and some concluding remarks are provided.
Journal of Statistical Theory and Practice, 2017
Under a convenient third-order framework, the asymptotic distributional behaviour of a class of l... more 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-bias tail index estimators with two classes of location invariant estimators of adequate second-order parameters, recently introduced in the literature. An application to simulated Student-t data and to the logexchange rates of the Euro against USA Dollar and Euro against GB Pound is also provided.
CIM Series in Mathematical Sciences, 2015
Resampling computer intensive methodologies, like the jackknife and the bootstrap are important t... more 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 parameter show the same type of behaviour: nice asymptotic properties, but a high variance for small k, the number of upper order statistics used in the estimation, a high bias for large k, and the need for an adequate choice of k. After a brief reference to some estimators of the aforementioned parameter and their asymptotic properties we present an algorithm that deals with an adaptive reliable estimation of ξ. Applications of these methodologies to the analysis of environmental and financial data sets are undertaken.
Journal of Statistical Theory and Practice, 2014
ABSTRACT For heavy right tails and under a semi-parametric framework, we introduce a class of loc... more ABSTRACT 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 class of estimators is achieved under a third-order condition on the right tail of the underlying model F for intermediate and large ranks, respectively. An illustration of the finite sample behaviour of the estimators is provided through a Monte-Carlo simulation study.
Advances in Theoretical and Applied Statistics, 2013
In this chapter, we consider a recent class of generalized negative moment estimators of a negati... more 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 makes them highly flexible and possibly second-order unbiased for a large variety of models. In this chapter, we are interested not only on the adaptive choice of the tuning parameters k and θ, but also on an application of these semi-parametric estimators to the analysis of sets of environmental and simulated data.
Communications in Statistics - Simulation and Computation, 2011
In this paper, we deal with semi-parametric corrected-bias estimation of a positive extreme value... more In this paper, we deal with semi-parametric corrected-bias estimation of a positive extreme value index (EVI). Then, the classical EVI-estimators are the Hill estimators, based on any intermediate number k of top order statistics. But these EVIestimators are not location-invariant, contrarily to the peaks over random threshold (PORT)-Hill estimators, which depend on an extra tuning parameter q. On the basis of second-order minimumvariance reduced-bias (MVRB) EVI-estimators, we shall here consider PORT-MVRB EVI-estimators, and propose the use of a heuristic algorithm, for the adaptive choice of k and q. Applications in the fields of insurance and finance will be provided. Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online.
Springer eBooks, Sep 13, 2012
In this chapter, we consider an application to environmental data of a bootstrap algorithm for th... more 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 from scale invariant, like the classical ones, are also location invariant. These estimators depend not only on an integer tuning parameter k, the number of top order statistics involved in the estimation, but also on an extra control real parameter q, 0 ≤ q < 1, which makes them highly flexible.
Communications in Statistics, Mar 24, 2009
Journal of The Royal Statistical Society Series B-statistical Methodology, Nov 2, 2007
We are interested in the derivation of the distributional properties of a weighted log-excesses e... more We are interested in the derivation of the distributional properties of a weighted log-excesses estimator of a positive tail index γ. One of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias, together with the maintenance of the asymptotic variance of the maximum likelihood estimator of γ, under a strict Pareto model. We consider the external estimation not only of a second-order shape parameter ρ but also of a second-order scale parameter β. This will enable us to reduce the asymptotic variance of the final estimators under consideration, compared with second-order reduced bias estimators that are already available in the literature. The second-order reduced bias estimators that are considered are also studied for finite samples, through Monte Carlo techniques, as well as applied to real data in the field of finance.
![Research paper thumbnail of Erratum to “Competitive estimation of the extreme value index” [Statist. Probab. Lett. 117 (2016) 128–135]](https://attachments.academia-assets.com/106414096/thumbnails/1.jpg)
](Statistics & Probability Letters, Nov 1, 2017
International Journal of Sports Science & Coaching, Oct 30, 2018
The International Swimming Federation has developed a points system that allows comparisons of re... more 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 correspondent event. Since it is based on only one observation, this work aims to suggest a new system, based on the probability distribution of the best performances in each event. Using extreme value theory, such distribution, under certain conditions, converges to a generalized Pareto distribution. The new performance index, based on the peaks over threshold methodology, is obtained based on the exceedance probabilities correspondent to the swimmers' times that exceed a given threshold. We work with 17 officially recognized events in 50 m pool, for each women and men, and considered all-time rankings for all events until 31 December 2016. A study on the adequacy of the proposed generalized Pareto distribution index and a comparison between the performances of Usain Bolt and Michael Phelps are also conducted.
Statistics & Probability Letters, Oct 1, 2016
The mean-of-order-p (MO p) extreme value index (EVI) estimators are based on Hölder's mean of an ... more 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. To make the MO p location-invariant, it is thus sensible to use the peaks over a random threshold (PORT) methodology, based upon the excesses over an adequate ascending order statistic. In this article, apart from an asymptotic comparison at optimal levels of the optimal MO p class and some competitive EVI-estimators, like a Pareto probability weighted moments EVI-estimator, a few details on PORT classes of EVI-estimators are provided, enhancing their high efficiency both asymptotically and for finite samples.
Journal of Statistical Computation and Simulation, Jun 1, 2010
John Wiley & Sons, Inc. eBooks, Oct 7, 2016
Birkhäuser Boston eBooks, 2008
Discussiones Mathematicae Probability and Statistics, 2010
In this article, we begin with an asymptotic comparison at optimal levels of the so-called "maxim... more In this article, we begin with an asymptotic comparison at optimal levels of the so-called "maximum likelihood" (ML) extreme value index estimator, based on the excesses over a high random threshold, denoted PORT-ML, with PORT standing for peaks over random thresholds, with a similar ML estimator, denoted PORT-MP, with MP standing for modified-Pareto. The PORT-MP estimator is based on the same excesses, but with a trial of accommodation of bias on the Generalized Pareto model underlying those excesses. We next compare the behaviour of these ML implicit estimators with the equivalent behaviour of a few explicit tail index estimators, the Hill, the moment, the generalized Hill and the mixed moment. As expected, none of the estimators can always dominate the alternatives, even when we include second-order MVRB tail index estimators, with MVRB standing for minimum-variance reduced-bias. However, the asymptotic performance of the MVRB estimators is quite interesting and provides a challenge for a further study of these MVRB estimators at optimal levels.
DOAJ (DOAJ: Directory of Open Access Journals), Dec 1, 2014
• In this paper we study, under a semi-parametric framework and for heavy right tails, a class of... more • 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 thresholds. Asymptotic normality of such estimators is achieved under a third-order condition on the right-tail of the underlying model F and for suitable large intermediate ranks. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte-Carlo simulation study.
Journal of the Royal Statistical Society, 2008
We are interested in the derivation of the distributional properties of a weighted log-excesses e... more We are interested in the derivation of the distributional properties of a weighted log-excesses estimator of a positive tail index γ. One of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias, together with the maintenance of the asymptotic variance of the maximum likelihood estimator of γ, under a strict Pareto model. We consider the external estimation not only of a second-order shape parameter ρ but also of a second-order scale parameter β. This will enable us to reduce the asymptotic variance of the final estimators under consideration, compared with second-order reduced bias estimators that are already available in the literature. The second-order reduced bias estimators that are considered are also studied for finite samples, through Monte Carlo techniques, as well as applied to real data in the field of finance.
Extremes, 2008
In statistics of extremes, inference is often based on the excesses over a high random threshold.... more In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called "maximum likelihood" estimators of the tail index γ . In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar "maximum likelihood" estimator of a positive tail index γ , based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study.
DOAJ (DOAJ: Directory of Open Access Journals), Jun 1, 2016
• A simple generalisation of the classical Hill estimator of a positive extreme value index (EVI)... more • 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 (MOP) of those statistics, with p real, and even an optimal MOP (OMOP) class of EVI-estimators. These estimators are scale invariant but not location invariant. With PORT standing for peaks over random threshold, new classes of PORT-MOP and PORT-OMOP EVI-estimators are now introduced. These classes are dependent on an extra tuning parameter q, 0 ≤ q < 1, and they are both location and scale invariant, a property also played by the EVI. The asymptotic normal behaviour of those PORT classes is derived. These EVI-estimators are further studied for finite samples, through a Monte-Carlo simulation study. An adequate choice of the tuning parameters under play is put forward, and some concluding remarks are provided.
Journal of Statistical Theory and Practice, 2017
Under a convenient third-order framework, the asymptotic distributional behaviour of a class of l... more 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-bias tail index estimators with two classes of location invariant estimators of adequate second-order parameters, recently introduced in the literature. An application to simulated Student-t data and to the logexchange rates of the Euro against USA Dollar and Euro against GB Pound is also provided.
CIM Series in Mathematical Sciences, 2015
Resampling computer intensive methodologies, like the jackknife and the bootstrap are important t... more 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 parameter show the same type of behaviour: nice asymptotic properties, but a high variance for small k, the number of upper order statistics used in the estimation, a high bias for large k, and the need for an adequate choice of k. After a brief reference to some estimators of the aforementioned parameter and their asymptotic properties we present an algorithm that deals with an adaptive reliable estimation of ξ. Applications of these methodologies to the analysis of environmental and financial data sets are undertaken.
Journal of Statistical Theory and Practice, 2014
ABSTRACT For heavy right tails and under a semi-parametric framework, we introduce a class of loc... more ABSTRACT 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 class of estimators is achieved under a third-order condition on the right tail of the underlying model F for intermediate and large ranks, respectively. An illustration of the finite sample behaviour of the estimators is provided through a Monte-Carlo simulation study.
Advances in Theoretical and Applied Statistics, 2013
In this chapter, we consider a recent class of generalized negative moment estimators of a negati... more 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 makes them highly flexible and possibly second-order unbiased for a large variety of models. In this chapter, we are interested not only on the adaptive choice of the tuning parameters k and θ, but also on an application of these semi-parametric estimators to the analysis of sets of environmental and simulated data.
Communications in Statistics - Simulation and Computation, 2011
In this paper, we deal with semi-parametric corrected-bias estimation of a positive extreme value... more In this paper, we deal with semi-parametric corrected-bias estimation of a positive extreme value index (EVI). Then, the classical EVI-estimators are the Hill estimators, based on any intermediate number k of top order statistics. But these EVIestimators are not location-invariant, contrarily to the peaks over random threshold (PORT)-Hill estimators, which depend on an extra tuning parameter q. On the basis of second-order minimumvariance reduced-bias (MVRB) EVI-estimators, we shall here consider PORT-MVRB EVI-estimators, and propose the use of a heuristic algorithm, for the adaptive choice of k and q. Applications in the fields of insurance and finance will be provided. Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online.
Springer eBooks, Sep 13, 2012
In this chapter, we consider an application to environmental data of a bootstrap algorithm for th... more 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 from scale invariant, like the classical ones, are also location invariant. These estimators depend not only on an integer tuning parameter k, the number of top order statistics involved in the estimation, but also on an extra control real parameter q, 0 ≤ q < 1, which makes them highly flexible.
Communications in Statistics, Mar 24, 2009
Journal of The Royal Statistical Society Series B-statistical Methodology, Nov 2, 2007
We are interested in the derivation of the distributional properties of a weighted log-excesses e... more We are interested in the derivation of the distributional properties of a weighted log-excesses estimator of a positive tail index γ. One of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias, together with the maintenance of the asymptotic variance of the maximum likelihood estimator of γ, under a strict Pareto model. We consider the external estimation not only of a second-order shape parameter ρ but also of a second-order scale parameter β. This will enable us to reduce the asymptotic variance of the final estimators under consideration, compared with second-order reduced bias estimators that are already available in the literature. The second-order reduced bias estimators that are considered are also studied for finite samples, through Monte Carlo techniques, as well as applied to real data in the field of finance.
Statistics & Probability Letters, Nov 1, 2017
International Journal of Sports Science & Coaching, Oct 30, 2018
The International Swimming Federation has developed a points system that allows comparisons of re... more 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 correspondent event. Since it is based on only one observation, this work aims to suggest a new system, based on the probability distribution of the best performances in each event. Using extreme value theory, such distribution, under certain conditions, converges to a generalized Pareto distribution. The new performance index, based on the peaks over threshold methodology, is obtained based on the exceedance probabilities correspondent to the swimmers' times that exceed a given threshold. We work with 17 officially recognized events in 50 m pool, for each women and men, and considered all-time rankings for all events until 31 December 2016. A study on the adequacy of the proposed generalized Pareto distribution index and a comparison between the performances of Usain Bolt and Michael Phelps are also conducted.
Statistics & Probability Letters, Oct 1, 2016
The mean-of-order-p (MO p) extreme value index (EVI) estimators are based on Hölder's mean of an ... more 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. To make the MO p location-invariant, it is thus sensible to use the peaks over a random threshold (PORT) methodology, based upon the excesses over an adequate ascending order statistic. In this article, apart from an asymptotic comparison at optimal levels of the optimal MO p class and some competitive EVI-estimators, like a Pareto probability weighted moments EVI-estimator, a few details on PORT classes of EVI-estimators are provided, enhancing their high efficiency both asymptotically and for finite samples.
Journal of Statistical Computation and Simulation, Jun 1, 2010
John Wiley & Sons, Inc. eBooks, Oct 7, 2016
Birkhäuser Boston eBooks, 2008
Discussiones Mathematicae Probability and Statistics, 2010
In this article, we begin with an asymptotic comparison at optimal levels of the so-called "maxim... more In this article, we begin with an asymptotic comparison at optimal levels of the so-called "maximum likelihood" (ML) extreme value index estimator, based on the excesses over a high random threshold, denoted PORT-ML, with PORT standing for peaks over random thresholds, with a similar ML estimator, denoted PORT-MP, with MP standing for modified-Pareto. The PORT-MP estimator is based on the same excesses, but with a trial of accommodation of bias on the Generalized Pareto model underlying those excesses. We next compare the behaviour of these ML implicit estimators with the equivalent behaviour of a few explicit tail index estimators, the Hill, the moment, the generalized Hill and the mixed moment. As expected, none of the estimators can always dominate the alternatives, even when we include second-order MVRB tail index estimators, with MVRB standing for minimum-variance reduced-bias. However, the asymptotic performance of the MVRB estimators is quite interesting and provides a challenge for a further study of these MVRB estimators at optimal levels.
DOAJ (DOAJ: Directory of Open Access Journals), Dec 1, 2014
• In this paper we study, under a semi-parametric framework and for heavy right tails, a class of... more • 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 thresholds. Asymptotic normality of such estimators is achieved under a third-order condition on the right-tail of the underlying model F and for suitable large intermediate ranks. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte-Carlo simulation study.
Journal of the Royal Statistical Society, 2008
We are interested in the derivation of the distributional properties of a weighted log-excesses e... more We are interested in the derivation of the distributional properties of a weighted log-excesses estimator of a positive tail index γ. One of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias, together with the maintenance of the asymptotic variance of the maximum likelihood estimator of γ, under a strict Pareto model. We consider the external estimation not only of a second-order shape parameter ρ but also of a second-order scale parameter β. This will enable us to reduce the asymptotic variance of the final estimators under consideration, compared with second-order reduced bias estimators that are already available in the literature. The second-order reduced bias estimators that are considered are also studied for finite samples, through Monte Carlo techniques, as well as applied to real data in the field of finance.