• The last decade has seen development of a plethora of approaches for threshold estimation in extreme value applications. From a statistical perspective, the threshold is loosely defined such that the population tail can be well approximated by an extreme value model (e.g., the generalised Pareto distribution), obtaining a balance between the bias due to the asymptotic tail approximation and parameter estimation uncertainty due to the inherent sparsity of threshold excess data. This paper reviews recent advances and some traditional approaches, focusing on those that provide quantification of the associated uncertainty on inferences (e.g., return level estimation).

• Classical extreme value index estimators are known to be quite sensitive to the number k of top order statistics used in the estimation. The recently developed second order reduced-bias estimators show much less sensitivity to changes in k. Here, we are interested in the improvement of the performance of reduced-bias extreme value index estimators based on an exponential second order regression model applied to the scaled log-spacings of the top k order statistics. In order to achieve that improvement, the estimation of a “scale” and a “shape” second order parameters in the bias is performed at a level k1 of a larger order than that of the level k at which we compute the extreme value index estimators. This enables us to keep the asymptotic variance of the new estimators of a positive extreme value index γ equal to the asymptotic variance of the Hill estimator, the maximum likelihood estimator of γ, under a strict Pareto model. These new estimators are then alternatives to the classical estimators, not only around optimal and/or large levels k, but for other levels too. To enhance the interesting performance of this type of estimators, we also consider the estimation of the “scale” second order parameter only, at the same level k used for the extreme value index estimation. The asymptotic distributional properties of the proposed class of γ-estimators are derived and the estimators are compared with other similar alternative estimators of γ recently introduced in the literature, not only asymptotically, but also for finite samples through Monte Carlo techniques. Case-studies in the fields of finance and insurance will illustrate the performance of the new second order reduced-bias extreme value index estimators.

† In this paper we are interested in an adequate estimation of the dominant component of the bias of Hill’s estimator of a positive tail index ∞, in order to remove it from the classical Hill estimator in difierent asymptotically equivalent ways. If the second order parameters in the bias are computed at an adequate level k1 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 ∞ 2 . The asymptotic distributional properties of the proposed estimators of ∞ are derived and the estimators are compared not only asymptotically, but also for flnite samples through Monte Carlo techniques.

type="main" xml:id="insr12058-abs-0001"> Statistical issues arising in modelling univariate extremes of a random sample have been successfully used in the most diverse fields, such as biometrics, finance, insurance and risk theory. Statistics of univariate extremes (SUE), the subject to be dealt with in this review paper, has recently faced a huge development, partially because rare events can have catastrophic consequences for human activities, through their impact on the natural and constructed environments. In the last decades, there has been a shift from the area of parametric SUE, based on probabilistic asymptotic results in extreme value theory, towards semi-parametric approaches. After a brief reference to Gumbel's block methodology and more recent improvements in the parametric framework, we present an overview of the developments on the estimation of parameters of extreme events and on the testing of extreme value conditions under a semi-parametric framework. We further discuss a few challenging topics in the area of SUE. © 2014 The Authors. International Statistical Review © 2014 International Statistical Institute

[1] Underestimation of extreme values is a widely acknowledged issue in daily precipitation simulation. Nonparametric precipitation generators have inherent limitations in representing extremes. Parametric generators can realistically model the full spectrum of precipitation amount through compound distributions. Nevertheless, fitting these distributions suffers from numerical instability, supervised learning, and computational demand. This study presents an easy-to-implement hybrid probability distribution to model the full spectrum of precipitation amount. The basic idea for the hybrid distribution lies in synthesizing low to moderate precipitation by an exponential distribution and extreme precipitation by a generalized Pareto distribution. By forcing the two distributions to be continuous at the junction point, the threshold of the generalized Pareto distribution can be implicitly learned in an unsupervised manner. Monte Carlo simulation shows that the hybrid distribution is capable of modeling heavy tailed data. Performance of the distribution is further evaluated using 49 daily precipitation records across Texas. Results show that the model is able to capture both the bulk and the tail of daily precipitation amount. The maximum goodness-of-fit and penalized maximum likelihood methods are found to be reliable complements to the maximum likelihood method, in that generally they can provide adequate goodness-of-fit. The proposed distribution can be incorporated into precipitation generators and downscaling models in order to realistically simulate the entire range of precipitation without losing extreme values.

We apply extreme value theory to assess the tail dependence between three currency crisis measures and 18 economic indicators commonly used for predicting crises. In our pooled sample of 46 countries in the period 1974–2008, we find that nearly all pairs of variables are asymptotically independent: in the limit, extreme values of economic indicators are not followed by severe currency crashes. Our findings may explain the poor performance of existing early warning systems for currency crises. However, we do find that economic variables with stronger extremal association with the exchange rate have better crisis prediction performance, both in-sample and out-of-sample.

We address the problem of estimating the Weibull tail-coefficient which is the regular variation exponent of the inverse failure rate function. We propose a family of estimators of this coefficient and an associate extreme quantile estimator. Their asymptotic normality are established and their asymptotic mean-square errors are compared. The results are illustrated on some finite sample situations

This paper is devoted to tail index estimation in the context of survey data. Assuming that the population of interest is described by a heavy-tailed statistical model, we prove that the survey scheme plays a crucial role in the design of consistent inference methods for extremes. As can be revealed by simulation experiments, ignoring the sampling plan generally induces a significant bias, jeopardizing the accuracy of the extreme value statistics thus computed. Focus is here on the celebrated Hill method for tail index estimation, it is shown how to modify it in order to take into account the survey design. Precisely, under specific conditions on the inclusion probabilities of first and second orders, we establish the consistency of the variant of the Hill estimator we propose. Additionally, its asymptotic normality is proved in a specific situation. Application of this limit result for building Gaussian confidence intervals is thoroughly discussed and illustrated by numerical results.

This article develops a threshold system for monitoring airline performance. This threshold system divides the sample space into regions with increasing levels of risk and allows instant assessments of risk level of any observed airline performance. Of particular concern is the performance with extreme risk. In this article, a multivariate extreme value theory approach is used to establish thresholds for signaling varying levels of extremeness in the context of simultaneous monitoring of multiple risk measures. The threshold system is justified in terms of multivariate extreme quantiles, and its sample estimator is shown to be consistent. This threshold system applies to general extreme risk management. Finally, a simulation and comparison study demonstrates the good performance of the proposed multivariate extreme quantile estimator. Supplemental materials providing technical details are available online.

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.

The main objective of statistics of univariate extremes lies in the estimation of quantities related to extreme events. In many areas of application, like finance, insurance and statistical quality control, a typical requirement is to estimate a high quantile, i.e. the Value at Risk at a level \(q (\)VaR\(_q)\), high enough, so that the chance of exceedance of that value is equal to \(q\), with \(q\) small. In this paper we deal with the semi-parametric estimation of VaR\(_q\), for heavy tails, introducing a new class of VaR-estimators based on a class of mean-of-order- \(p\) (MOP) extreme value index (EVI)-estimators, recently introduced in the literature. Interestingly, the MOP EVI-estimators can have a mean square error smaller than that of the classical EVI-estimators, even for small values of \(k\). They are thus a nice basis to build alternative VaR-estimators not only around optimal levels, but for other levels too.The new VaR-estimators are compared with the classical ones, not only asymptotically, but also for finite samples, through Monte-Carlo techniques.

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 estimate as a function of k. Some of these procedures will be reviewed. Despite of the fact 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 largest possible earthquake magnitude based on geographical characteristics for a selected return period is required in earthquake engineering, disaster management, and insurance. Ground-based observations combined with statistical analyses may offer new insights into earthquake prediction. In this study, to investigate the seismic characteristics of different geographical regions in detail, clustering was used to provide earthquake zoning for Mainland China based on the geographical features of earthquake events. In combination with geospatial methods, statistical extreme value models and the right-truncated Gutenberg–Richter model were used to analyze the earthquake magnitudes of Mainland China under both clustering and non-clustering. The results demonstrate that the right-truncated peaks-over-threshold model is the relatively optimal statistical model compared with classical extreme value theory models, the estimated return level of which is very close to that of the geographical-based right-truncated Gutenberg–Richter model. Such statistical models can provide a quantitative analysis of the probability of future earthquake risks in China, and geographical information can be integrated to locate the earthquake risk accurately.

The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. This paper proposes a new estimator for high quantiles based on the geometric mean. The new estimator has good asymptotic properties as well as it provides a computational algorithm for estimating confidence intervals of high quantiles. The new estimator avoids difficulties, improves efficiency and reduces bias. Comparisons of efficiencies and biases of the new estimator relative to existing estimators are studied. The theoretical are confirmed through Monte Carlo simulations. Finally, the applications on two real-world examples are provided.

The estimation of the tail index is a very important issue within extreme value theory. Semi-parametric estimators usually require the choice of the number k of upper order statistics to use in the estimation, which is a difficult problem to handle. Here it is applied a heuristic graphical method to well-known semi-parametric tail index estimators and analyze it through simulation. It will be seen that some criteria lead to good results.

The classical Hill estimator of a positive extreme value index (EVI) can be regarded as the logarithm of the geometric mean, or equivalently the logarithm of the mean of order p = 0 , of a set of adequate statistics. A simple generalisation of the Hill estimator is now proposed, considering a more general mean of order p ? 0 of the same statistics. Apart from the derivation of the asymptotic behaviour of this new class of EVI-estimators, an asymptotic comparison, at optimal levels, of the members of such class and other known EVI-estimators is undertaken. An algorithm for an adaptive estimation of the tuning parameters under play is also provided. A large-scale simulation study and an application to simulated and real data are developed.

The authors examine the asymptotic behaviour of conditional threshold exceedance probabilities for an elliptically distributed pair (X, Y) of random variables. More precisely, they investigate the limiting behaviour of the conditional distribution of Y given that X becomes extreme. They show that this behaviour differs between regularly and rapidly varying tails. Le comportement extreme des lois elliptiques bivariees: Les auteurs s'interessent au comportement asymptotique de probabilites conditionnelles de depassement d'un seuil pour une paire (X, Y) de variables aleatoires de loi elliptique. Plus precisement, ils etudient le comportement limite de la loi conditionnelle de Y sachant que X devient extrěme. ns montrent que ce comportement difere suivant que les queues de la loi sont a variations regulieres ou rapides.

The appropriate choice of a threshold level, which separates the tails of the probability distribution of a random variable from its middle part, is considered to be a very complex and challenging task. This paper provides an empirical study on various methods of the optimal tail selection in risk measurement. The results indicate which method may be useful in practice for investors and financial and regulatory institutions. Some methods that perform well in simulation studies, based on theoretical distributions, may not perform well when real data are in use. We analyze twelve methods with different parameters for forty-eight world indices using returns from the period of 2000–Q1 2020 and four sub-periods. The research objective is to compare the methods and to identify those which can be recognized as useful in risk measurement. The results suggest that only four tail selection methods, i.e., the Path Stability algorithm, the minimization of the Asymptotic Mean Squared Error approach, the automated Eyeball method with carefully selected tuning parameters and the Hall single bootstrap procedure may be useful in practical applications.

The aim of this paper is to give a general model for predicting fatigue behavior for any stress level and range based on laboratory tests. More precisely, an existing Weibull model is generalized and adapted to this type of predictions via compatibility and functional equations techniques. A wide family of models is obtained, for which inference, testing hypotheses, model choice and model validation problems are dealt with. Laboratory testing strategies adequate to the estimation process are also briefly discussed. The relevant result is that testing four groups of specimens at four different stress levels is sufficient to estimate the parameters of the model and consequently, to give the Wohler fields for any stress level. Together the proposed model and laboratory tests allow any fatigue analysis to be performed. This includes a possible solution of the fatigue damage accumulation problem. One example of application is given to illustrate the proposed methods.

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.