Minimum-variance Reduced-bias Research Articles

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.

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.

A new partially reduced-bias mean-of-order [formula omitted] class of extreme value index estimators

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.

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.

Generalized Jackknife-Based Estimators for Univariate Extreme-Value Modeling

In this article, we revisit the importance of the generalized jackknife in the construction of reliable semi-parametric estimates of some parameters of extreme or even rare events. The generalized jackknife statistic is applied to a minimum-variance reduced-bias estimator of a positive extreme value index—a primary parameter in statistics of extremes. A couple of refinements are proposed and a simulation study shows that these are able to achieve a lower mean square error. A real data illustration is also 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.

Adaptive Reduced-Bias Tail Index and VaR Estimation via the Bootstrap Methodology

In this article, we deal with the estimation, under a semi-parametric framework, of a positive extreme value index γ, the primary parameter in Statistics of Extremes, and associated estimation of the Value at Risk (VaR) at a level p, the size of the loss occurred with a small probability p. We consider second-order minimum-variance reduced-bias (MVRB) estimators, and propose the use of bootstrap computer-intensive methods for the adaptive choice of thresholds, both for γ and Var p . Applications in the fields of insurance and finance, as well as a small-scale simulation study of the bootstrap adaptive estimators’ behaviour, are also 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.

Comparison at optimal levels of classical tail index estimators: a challenge for reduced-bias estimation?

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.

Reduced-Bias Tail Index Estimators Under a Third-Order Framework

In this article, 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 γ. 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.

A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator

For heavy tails, with a positive tail index γ , classical tail index estimators, like the Hill estimator, are known to be quite sensitive to the number of top order statistics k used in the estimation, whereas second-order reduced-bias estimators show much less sensitivity to changes in k . In the recent minimum-variance reduced-bias (MVRB) tail index estimators, the estimation of the second order parameters in the bias has been performed at a level k 1 of a larger order than that of the level k at which we compute the tail index estimators. Such a procedure enables us to keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator, for all k at which we can guarantee the asymptotic normality of the Hill statistics. These values of k , as well as larger values of k , will also enable us to guarantee the asymptotic normality of the reduced-bias estimators, but, to reach the minimal mean squared error of these MVRB estimators, we need to work with levels k and k 1 of the same order. In this note we derive the way the asymptotic variance varies as a function of q , the finite limiting value of k / k 1 , as the sample size n increases to infinity.

PORT Hill and Moment Estimators for Heavy-Tailed Models

In this article, we use the peaks over random threshold (PORT)-methodology, and consider Hill and moment PORT-classes of extreme value index estimators. These classes of estimators are invariant not only to changes in scale, like the classical Hill and moment estimators, but also to changes in location. They are based on the sample of excesses over a random threshold, the order statistic X [np]+1:n , 0 ≤ p < 1, being p a tuning parameter, which makes them highly flexible. Under convenient restrictions on the underlying model, these classes of estimators are consistent and asymptotically normal for adequate values of k, the number of top order statistics used in the semi-parametric estimation of the extreme value index γ. In practice, there may however appear a stability around a value distant from the target γ when the minimum is chosen for the random threshold, and attention is drawn for the danger of transforming the original data through the subtraction of the minimum. A new bias-corrected moment estimator is also introduced. The exact performance of the new extreme value index PORT-estimators is compared, through a large-scale Monte-Carlo simulation study, with the original Hill and moment estimators, the bias-corrected moment estimator, and one of the minimum-variance reduced-bias (MVRB) extreme value index estimators recently introduced in the literature. As an empirical example we estimate the tail index associated to a set of real data from the field of finance.