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The role of generalized means in multivariate extreme value statistics

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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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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.
Book
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.