Statistics of Multivariate Extremes (original) (raw)

An Overview and Open Research Topics in Statistics of Univariate Extremes

2012

• This review paper focuses on statistical issues arising in modeling univariate extremes of a random sample. In the last three decades there has been a shift from the area of parametric statistics of extremes, based on probabilistic asymptotic results in extreme value theory, towards a semi-parametric approach, where the estimation of the right and/or left tail-weight is performed under a quite general framework. But new parametric models can still be of high interest for the analysis of extreme events, if associated with appropriate statistical inference methodologies. After a brief reference to Gumbel's classical block methodology and later improvements in the parametric framework, we present an overview of the developments on the estimation of parameters of extreme events and testing of extreme value conditions under a semi-parametric framework, and discuss a few challenging open research topics.

Extreme Value Theory and Statistics of Univariate Extremes: A Review

International Statistical Review, 2014

Statistical issues arising in modeling 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 due to the fact that 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.

Some contributions in probability and statistics of extremes

A mes co-auteurs, Jürg Hüsler et Holger Rootzén rencontrés au Center for Stochastic Processesà Chapel Hill lors de ma thèse, Sid Resnick qui m'a prise en post-doctoratà Cornell et que je remercie encore, José León dit Chichi, Pierre Picco, Miguel Atencia et Gonzalo Joya, avec lesquels ce fut et c'est un réel plaisir de travailler et de partager toutes les joies et les peines liéesà l'activité de recherche. A Chichi et Bernard Bru, que j'estime tant pour leurs qualités humaines que scientifiques, et dont l'exemple est pour moi un formidable moteur. Encore un grand merci ! Aux membres du jury : Philippe Soulier et Mario Wschebor, rapporteurs choisis par la commission parisienne des thèses, Xavier Guyon, qui a présenté mon dossierà Paris 1, Jean-Marc Azaïs, Yuri Davydov, Anne Estrade et Annie Millet, que je remercie chaleureusement de m'avoir fait l'honneur de participerà mon jury.

Multivariate extremes based on a notion of radius

2015

Modeling and understanding multivariate extreme events is challenging, but of great importance invarious applications— e.g. in biostatistics, climatology, and finance. The separating Hill estimator canbe used in estimating the extreme value index of a heavy tailed multivariate elliptical distribution. Weconsider the asymptotic behavior of the separating Hill estimator under estimated location and scatter.The asymptotic properties of the separating Hill estimator are known under elliptical distribution withknown location and scatter. However, the effect of estimation of the location and scatter has previouslybeen examined only in a simulation study. We show, analytically, that the separating Hill estimator isconsistent and asymptotically normal under estimated location and scatter, when certain mild conditionsare met.

Ordered multivariate extremes

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 1998

Multivariate extreme value models and associated statistical methods are developed for vector observations whose components are subject to an order restriction. The approach extends the multivariate threshold methodology of Coles and Tawn, Joe and co-workers and Smith and co-workers. The results are illustrated by an analysis of extreme rainfalls of different durations, and by a study of the problem of linking a long series of daily rainfall extremes with a partially overlapping shorter series of hourly extremes.

Extremes for multivariate expectiles

Statistics & Risk Modeling, 2018

Multivariate expectiles, a new family of vector-valued risk measures, were recently introduced in the literature. [22]. Here we investigate the asymptotic behavior of these measures in a multivariate regular variation context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles in the Fréchet domain of attraction case with asymptotic independence, or for comonotonic marginal distributions.

Extremality in multivariate statistics

2012

Quiero expresar especial agradecimiento al Profesor Franco Pellerey del Politecnico di Torino por su excelente asesoría, dedicación y hospitalidad en mi visita a Turín. Al Profesor Froilan Martínez por su valiosa tutoría sobre rotaciones, a Ignacio Cascos por sus oportunas tutorias sobre riesgos. A todas las personas nombradas muchas gracias.

Statistics of Bivariate Extreme Values

2014

LIST OF FIGURES 2.11 (Left) The density of the Frank copula with a = 0:5: (Rihgt) Estimation of the copula density using a Gaussian kernel and Gaussian transformations with 1,000 observations drawn from the Frank copula. 69 2.12 Dependogrammes for simulated data from three di¤erent copulas. .. 70 2.13 Kendall plots comparing a sample of simulated data from a Student copula (correlation 0.5, 3 degree of freedom) several copulas estimated on the same sample. .

Dependence of Multivariate Extremes

Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications, 2013

We give necessary and sufficient conditions for two sub-vectors of a random vector with a multivariate extreme value distribution, corresponding to the limit distribution of the maximum of a multidimensional stationary sequence with extremal index, to be independent or totally dependent. Those conditions involve first relations between the multivariate extremal indexes of the sequences and secondly a coefficient that measure the strength of dependence between both sub-vectors. The main results are illustrated with an auto-regressive sequence and a 3-dependent sequence.