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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 [ABSTRACT FROM AUTHOR]

The paper focuses primarily on temperature extremes measured at 24 European stations with at least 90 years of data. Here, the term extremes refers to rare excesses of daily maxima and minima. As mean temperatures in this region have been warming over the last century, it is automatic that this positive shift can be detected also in extremes. After removing this warming trend, we focus on the question of determining whether other changes are still detectable in such extreme events. As we do not want to hypothesize any parametric form of such possible changes, we propose a new non-parametric estimator based on the Kullback-Leibler divergence tailored for extreme events. The properties of our estimator are studied theoretically and tested with a simulation study. Our approach is also applied to seasonal extremes of daily maxima and minima for our 24 selected stations. [ABSTRACT FROM AUTHOR]

We present families of nonparametric estimators for the conditional tail index of a Pareto-type distribution in the presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second-order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment. [ABSTRACT FROM PUBLISHER]

. We introduce and study a class of weighted functional estimators for the coefficient of tail dependence in bivariate extreme value statistics. Asymptotic normality of these estimators is established under a second-order condition on the joint tail behaviour, some conditions on the weight function and for appropriately chosen sequences of intermediate order statistics. Asymptotically unbiased estimators are constructed by judiciously chosen linear combinations of weighted functional estimators, and variance optimality within this class of asymptotically unbiased estimators is discussed. The finite sample performance of some specific examples from our class of estimators and some alternatives from the recent literature are evaluated with a small simulation experiment. [ABSTRACT FROM AUTHOR]

We consider the estimation of the scale parameter appearing in the second order condition when the distribution underlying the data is of Pareto-type. Inspired by the work of Goegebeur et al. (J Stat Plan Inference 140:2632-2652, ) on the estimation of the second order rate parameter, we introduce a flexible class of estimators for the second order scale parameter, which has weighted sums of scaled log spacings of successive order statistics as basic building blocks. Under the second order condition, some conditions on the weight functions, and for appropriately chosen sequences of intermediate order statistics, we establish the consistency of our class of estimators. Asymptotic normality is achieved under a further condition on the tail function 1 − F, the so-called third order condition. As the proposed estimator depends on the second order rate parameter, we also examine the effect of replacing the latter by a consistent estimator. The asymptotic performance of some specific examples of our proposed class of estimators is illustrated numerically, and their finite sample behavior is examined by a small simulation experiment. [ABSTRACT FROM AUTHOR]

Interest is centered on the maximum likelihood (ML) estimators of the parameters of the Generalized Pareto Distribution in an extreme value context. Our aim consists of reducing the bias of these estimates for which no explicit expression is available. To circumvent this difficulty, we prove that these estimators are asymptotically equivalent to one-step estimators introduced by Beirlant et al. (2010) in a right-censoring context. Then, using this equivalence property, we estimate the bias of these one-step estimators to approximate the asymptotic bias of the ML-estimators. Finally, a small simulation study and an application to a real data set are provided to illustrate that these new estimators actually exhibit reduced bias. [ABSTRACT FROM AUTHOR]

Abstract: In the process of analyzing data, testing the fit of a model under consideration is a prerequisite for performing inference about the model parameters. In this paper we examine the goodness-of-fit testing problem for assessing whether a sample is consistent with the Weibull-type model. Inspired by the Jackson and the Lewis test statistics, originally proposed as goodness-of-fit tests for the exponential distribution, we introduce two new statistics for testing Weibull-type behavior, and study their asymptotic properties. Moreover, given that the statistics are ratios of estimators for the Weibull-tail coefficient, we obtain new estimators for the latter, and establish their consistency and asymptotic normality. The small sample behavior of our statistics and estimators is evaluated on the basis of a simulation study. [Copyright &y& Elsevier]

A new approach is suggested for choosing the threshold when fitting the Hill estimator of a tail exponent to extreme value data. Our method is based on an easily computed diagnostic, which in turn is founded directly on the Hill estimator itself, 'symmetrized' to remove the effect of the tail exponent but designed to emphasize biases in estimates of that exponent. The attractions of the method are its accuracy, its simplicity and the generality with which it applies. This generality implies that the technique has somewhat different goals from more conventional approaches, which are designed to accommodate the minor component of a postulated two-component Pareto mixture. Our approach does not rely on the second component being Pareto distributed. Nevertheless, in the conventional setting it performs competitively with recently proposed methods, and in more general cases it achieves optimal rates of convergence. A by-product of our development is a very simple and practicable exponential approximation to the distribution of the Hill estimator under departures from the Pareto distribution. [ABSTRACT FROM AUTHOR]

Abstract: In climatology and hydrology, univariate Extreme Value Theory has become a powerful tool to model the distribution of extreme events. The Generalized Pareto Distribution (GPD) is routinely applied to model excesses in space or time by letting the two GPD parameters depend on appropriate covariates. Two possible pitfalls of this strategy are the modeling and the interpretation of the scale and shape GPD parameters estimates which are often and incorrectly viewed as independent variables. In this note we first recall a statistical technique that makes the GPD estimates less correlated within a Maximum Likelihood (ML) estimation approach. In a second step we propose novel reparametrizations for two method-of-moments particularly popular in hydrology: the Probability Weighted Moment (PWM) method and its generalized version (GPWM). Finally these three inference methods (ML, PWM and GPWM) are compared and discussed with respect to the issue of correlations. [Copyright &y& Elsevier]

Abstract: In , a new family of distributions is introduced, depending on two parameters and , which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for and extreme quantiles are also proposed, but they both depend on the unknown parameter , making them useless in practical situations. In this paper, we propose an estimator of which is independent of . Plugging our estimator of in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set. [Copyright &y& Elsevier]