Lígia Henriques-Rodrigues

Adaptive PORT–MVRB estimation: an empirical comparison of two heuristic algorithms

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 location and scale-invariant estimators, based on the PORT methodology applied to second-order MVRB EVI-estimators. Trivial adaptations of these algorithms make them work for a similar estimation of other parameters of extreme events, such as the Value-at-Risk at a level p, the expected shortfall and the probability of exceedance of a high level x, among others. Applications to simulated data sets and to real data sets in the field of finance are provided.

Reduced-Bias Location-Invariant Extreme Value Index Estimation: A Simulation Study

In this article, we deal with semi-parametric corrected-bias estimation of a positive extreme value index (EVI), the primary parameter in statistics of extremes. Under such a context, the classical EVI-estimators are the Hill estimators, based on any intermediate number k of top-order statistics. But these EVI-estimators are not location-invariant, contrarily to the PORT-Hill estimators, which depend on an extra tuning parameter q, with 0 ≤ q < 1, and where PORT stands for peaks over random threshold. On the basis of second-order minimum-variance reduced-bias (MVRB) EVI-estimators, we shall here consider PORT-MVRB EVI-estimators. Due to the stability on k of the MVRB EVI-estimates, we propose the use of a heuristic algorithm, for the adaptive choice of k and q, based on the bias pattern of the estimators as a function of k. Applications in the fields of insurance and finance will be provided.

A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation

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 not enjoy the adequate linear property of quantiles, contrarily to the PORT VaR estimators, which depend on an extra tuning parameter q, with 0≤q<1. We shall here consider ‘quasi-PORT’ reduced-bias VaR estimators, for which such a linear property is obtained approximately. They are based on a partially shifted version of a minimum-variance reduced-bias (MVRB) estimator of the extreme value index (EVI), the primary parameter in Statistics of Extremes. Due to the stability on k of the MVRB EVI and associated VaR estimates, we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks.

A location-invariant probability weighted moment estimation of the Extreme Value Index

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.

Refined Estimation of a Light Tail: An Application to Environmental Data

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.

A Note on the Port Methodology in the Estimation of a Shape Second-Order Parameter

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 thresholds. Consistency of such estimators is achieved under a second-order condition on the right-tail of the underlying model F and for large intermediate ranks.

Adaptive PORT-MVRB Estimation of the Extreme Value Index

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.

Location-invariant reduced-bias tail index estimation under a third-order framework

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-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 log-exchange rates of the Euro against the U.S. dollar and the Euro against the GB pound is also provided.

Resampling-Based Methodologies in Statistics of Extremes: Environmental and Financial Applications

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.

Peaks Over Random Threshold Asymptotically Best Linear Estimation of the Extreme Value Index

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 parameters under play is proposed and applied to a set of financial data.