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Modern risk assessment of the amount of tail dependence is nowadays crucial in the most diverse fields, like finance and insurance, among others, and the correlation structure is usually not enough to describe such a tail dependence. Generalized means (GMs) are now used under a multivariate framework, essentially for the estimation of the tail dependence coefficient, in bivariate extreme value statistics. Associated asymptotically unbiased estimators are also constructed. The finite-sample behavior, as well as robustness, regarding sensitivity to the extreme value dependence assumption, is assessed through a small-scale Monte-Carlo simulation study.

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The Lehmer mean of order p of k positive numbers generalizes both the arithmetic mean (p = 1) and the harmonic mean (p = 0). Given a random sample and the associated sample of ascending order statistics, the classical Hill estimator of a positive extreme value index (EVI), the primary parameter of extreme events, can thus be considered as the Lehmer mean of order 1 of the k log-excesses. We now more generally consider the Lehmer mean of order p of the log-excesses and an associated Lehmer EVI-estimator. Apart from the derivation of the asymptotic behaviour of this class of EVI-estimators, an asymptotic comparison, at optimal levels, of the members of such a class reveals that for the optimal p they are able to overall outperform a recent and promising generalization of the Hill EVI-estimator. A large-scale Monte-Carlo simulation study is developed, giving emphasis to the discrepancies between asymptotic and finite sample behaviour of the estimators. A bootstrap algorithm for an adaptive estimation of the tuning parameters under play is also put forward.

Under a bivariate normal set-up, when we face a sample of maximum values and their concomitants, the estimation of regression coefficients by least squares, together with the normality of concomitants, enables us to derive a consistent estimator of the correlation coefficient of the original bivariate observations. In this paper we shall also consider the most usual multivariate ordering in Statistical Theory of Extreme Values, where
the original observations are destroyed, and we consider both maximum values of observed components. A second estimator of correlation coefficient is then proposed. The asymptotic and finite sample properties of both estimators are derived. They are then compared to the usual estimator of serial correlation.

Lehmer's mean-of-order p ($L_p$) generalizes the arithmetic mean, and $L_p$ extreme value index (EVI)-estimators can be easily built, as a generalization of the classical Hill EVI-estimators. Apart from a reference to the asymptotic behaviour of this class of estimators, an asymptotic comparison, at optimal levels, of the members of such a class reveals that for the optimal (p, k) in the sense of minimal mean square error, with k the number of top order statistics involved in the estimation, they are able to overall outperform a recent and promising generalization of the Hill EVI-estimator, related to the power mean, also known as Hölder's mean-of-order-p. A further comparison with other 'classical' non-reduced-bias estimators still reveals the competitiveness of this class of EVI-estimators.

When modeling extreme events, there are a few primordial parameters, among which we refer to the extreme value index (EVI) and the extremal index (EI). Under a framework related to large values, the EVI measures the right tail weight of the underlying distribution and the EI characterizes the degree of local dependence in the extremes of a stationary sequence. Most of the semiparametric estimators of these parameters show the same type of behavior: nice asymptotic properties but a high variance for small values of k, the number of upper order statistics used in the estimation, and a high bias for large values of k. This brings a real need for the choice of k. Choosing some well-known estimators of those two parameters, we revisit the application of a heuristic algorithm for the adaptive choice of k. A simulation study illustrates the performance of the proposed algorithm.

O Jackknife e o Bootstrap são metodologias de re-amostragem que têm frequentemente respondido pela afirmativa à questão de como pode a combinação de informação melhorar a qualidade de estimadores de determinado parâmetro ou funcional. Tais metodologias começaram recentemente a ser utilizadas em Estatística de Extremos, na estimação de parâmetros de acontecimentos raros, como o índice de cauda, intimamente relacionado com o peso da cauda direita de um modelo, ou o índice extremal, que pode ser informalmente definido como o recíproco do tempo médio de duração de acontecimentos extremos. Iremos neste trabalho referir alguns aspectos em que nos parece ser relevante em Estatística de Extremos a utilização das referidas metodologias de re-amostragem.

The extreme value index (EVI) is the primary parameter of extreme events. The EVI is used to characterize the tail behavior of a distribution, and it helps to indicate the size and frequency of certain extreme events under a given probability model: for large events, the bigger the EVI is, the heavier is the right-tail of the underlying parent distribution. The Lehmer mean of order p of the k log-excesses over the k + 1-th upper order statistic has been recently considered in the literature for the estimation of a positive EVI, associated with large extreme events. Such a Lehmer mean of order p generalizes the arithmetic mean (p = 1), the classical Hill estimator of a positive EVI, and for p > 1 has revealed to be very competitive for small values of the EVI, comparing favorably with one the simplest classes of reduced-bias EVI-estimators, a corrected-Hill estimator. Now, the comparison to other EVI-estimators is performed, and some information on the robustness of such a general class is provided, including its resistance to possible contamination by outliers.

After outlining the most importante results of classical probabilitic extreme value theory ando f its extensions for several forms of weak dependence, we mention Gumbel’s approach to statistical inference using extrema, and we point out some of its drawbacks: wasting of information by only considering the maxima of groups of observation, though generally records of the top order statistics are available; arbitrary grouping of data; extreme slowness of convergence towards the asymptotic law for some parente distributions. Concerning a more eficiente use of the information available, we obtain the asymptotic form of the joint distribution of the largest $i$ order statistics, $i$ a fixed integer. We then derive distributional properties of particular functions of a vector with such a distribution and develop several estimation techniques for dealing with multivariate samples of such independent vectors, namely by considering the order statistics of their first componentes and their concomitants. We next sketch an approach to extreme value practice, similar in spirit to Pickands’, and which avoids the arbitrary grouping of data. We also extend the results formerly obtained to the case of weakly dependente variables. Regarding the third drawback we mention, we consider a broad class of distributions $F$ in the domain of attraction of the type I extreme value distribution $\Lambda$ such that the convergence of $F^n (a_n x+b_n)$ (for suitable attraction coefficients $a_n$ and $b_n$) is very slow, and show that for members of this class there is, even for moderately large $n$, either a type II or type III distribution which is closer to the actual $F^n$ than $\Lambda$. This amplifies Fisher and Tippet’s result about the ``penultimate’’ behaviour of the maximum of independente normal variables. Finally, using simulation methos, we try to assess the advantages of fitting such a penultimate rathen than an ultimate approximation.

The statistical analysis of extreme data is important for various disciplines, including hydrology, insurance, finance, engineering and environmental sciences. This book provides a self-contained introduction to the parametric modeling, exploratory analysis and statistical interference for extreme values.
The entire text of this third edition has been thoroughly updated and rearranged to meet the new requirements. Additional sections and chapters, elaborated on more than 100 pages, are particularly concerned with topics like dependencies, the conditional analysis and the multivariate modeling of extreme data. Parts I–III about the basic extreme value methodology remain unchanged to some larger extent, yet notable are, e.g., the new sections about "An Overview of Reduced-Bias Estimation" (co-authored by M.I. Gomes), "The Spectral Decomposition Methodology", and "About Tail Independence" (co-authored by M. Frick), and the new chapter about "Extreme Value Statistics of Dependent Random Variables" (co-authored by H. Drees). Other new topics, e.g., a chapter about "Environmental Sciences", (co--authored by R.W. Katz), are collected within Parts IV–VI.