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REVSTAT No 3 ISSN 1645-6726 Instituto Nacional de Estatística Statistics Portugal REVSTAT Statistical Journal Special issue on “Collection of Surveys on Tail Event Modeling" Guest Editors: Miguel de Carvalho Anthony C. Davison Jan Beirlant Volume 10, No.1 March 2012 REVSTAT STATISTICAL JOURNAL Catalogação Recomendada REVSTAT. Lisboa, 2003- Revstat : statistical journal / ed. Instituto Nacional de Estatística. - Vol. 1, 2003- . - Lisboa I.N.E., 2003- . - 30 cm Semestral. - Continuação de : Revista de Estatística = ISSN 0873-4275. - edição exclusivamente em inglês ISSN 1645-6726 CREDITS - EDITOR-IN-CHIEF - PUBLISHER - M. Ivette Gomes - Instituto Nacional de Estatística, I.P. (INE, I.P.) Av. António José de Almeida, 2 - CO-EDITOR 1000-043 LISBOA - M. Antónia Amaral Turkman PORTUGAL - ASSOCIATE EDITORS Tel.: + 351 218 426 100 - Barry Arnold Fax: + 351 218 426 364 - Helena Bacelar- Nicolau Web site: http://www.ine.pt - Susie Bayarri Customer Support Service - João Branco (National network): 808 201 808 - M. Lucília Carvalho (Other networks): + 351 226 050 748 - David Cox - COVER DESIGN - Edwin Diday - Mário Bouçadas, designed on the stain glass - Dani Gamerman window at INE, I.P., by the painter Abel Manta - Marie Husková - Isaac Meilijson - LAYOUT AND GRAPHIC DESIGN - M. Nazaré Mendes-Lopes - Carlos Perpétuo - Stephan Morgenthaler - PRINTING - António Pacheco - Instituto Nacional de Estatística, I.P. - Dinis Pestana - Ludger Rüschendorf - EDITION - Gilbert Saporta - 300 copies - Jef Teugels - LEGAL DEPOSIT REGISTRATION - EXECUTIVE EDITOR - N.º 191915/03 - Maria José Carrilho - SECRETARY - Liliana Martins PRICE [VAT included] - Single issue ……………………………………………………….. € 11 - Annual subscription (No. 1 Special Issue, No. 2 and No.3)………. € 26 - Annual subscription (No. 2, No. 3) ……………………………….. € 18 © INE, I.P., Lisbon. Portugal, 2012* Reproduction authorised except for commercial purposes by indicating the source. FOREWORD Modern methods of statistics of extremes have been widely applied in risk modeling, with more and more scientists concerned with extrapolating to the tails of a distribution, often beyond any existing data. Complex estimation and inference problems arise when assessing the probability of such rare events, and ingenious statistical methods have been developed based on various assumptions useful for extrapolation. This special issue of Revstat—Statistical Journal presents an overview of such methods, by discussing recent developments in the statistical modeling of extremes and their applications to the analysis of risk. It covers a variety of topics, from methods for scalar extremes, often applied in a discrete time setting, to the infinite-dimensional setting, thus far mostly applied in the space domain. This special issue aims to provide a broad view of such topics, integrating modern advances with the historical perspective. This allows the authors to go to the origins of the concepts, methods and models of extremes, but in the light of the current state of the art. A Collection of Surveys on Tail Event Modeling tries, however, to offer more, as some challenges for future work are pinpointed by the authors. We hope these papers stimulate interaction between experts in the field of extremes and that they are useful for those entering this important field. We thank Ivette Gomes, editor-in-chief of Revstat—Statistical Journal, for encouraging us to take on this challenge. On the behalf of the Editorial Board we would like to thank the authors for contributing to this special issue. M. B. de Carvalho and A. C. Davison J. Beirlant Ecole Polytechnique Fed´ erale´ de Lausanne University of Leuven Institute of Mathematics Katholieke Universiteit Leuven Lausanne Leuven Switzerland Belgium INDEX An Overview and Open Research Topics in Statistics of Univariate Extremes Jan Beirlant, Frederico Caeiro and M. Ivette Gomes ............... 1 A Review of Extreme Value Threshold Estimation and Uncertainty Quantification Carl Scarrott and Anna MacDonald ................................ 33 Max-Stable Models for Multivariate Extremes Johan Segers ......................................................... 61 Bivariate Extreme Statistics, II Miguel de Carvalho and Alexandra Ramos .......................... 83 Modelling Time Series Extremes V. Chavez-Demoulin and A.C. Davison ............................. 109 A Survey of Spatial Extremes: Measuring Spatial Dependence and Modeling Spatial Effects Daniel Cooley, Jessi Cisewski, Robert J. Erhardt, Soyoung Jeon, Elizabeth Mannshardt, Bernard Oguna Omolo and Ying Sun . 135 Abstracted/indexed in: Current Index to Statistics, DOAJ, Google Scholar, Journal Citation Reports/Science Edition, Mathematical Reviews, Science Citation Index Expandedr, SCOPUS and Zentralblatt fur¨ Mathematic. REVSTAT – Statistical Journal Volume 10, Number 1, March 2012, 1–31 AN OVERVIEW AND OPEN RESEARCH TOPICS IN STATISTICS OF UNIVARIATE EXTREMES Authors: Jan Beirlant – Catholic University of Leuven, Belgium [email protected] Frederico Caeiro – New University of Lisbon, F.C.T. and C.M.A., Portugal [email protected] M. Ivette Gomes – University of Lisbon, F.C.U.L. (D.E.I.O.) and C.E.A.U.L., Portugal [email protected] Abstract: 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 ex- treme 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 parame- ters of extreme events and testing of extreme value conditions under a semi-parametric framework, and discuss a few challenging open research topics. Key-Words: extreme value index; parameters of extreme events; parametric and semi-parametric • estimation and testing; statistics of univariate extremes. AMS Subject Classification: 62G32, 62E20. • 2 J. Beirlant, Frederico Caeiro and M. Ivette Gomes An Overview and Open Research Topics in Statistics of Univariate Extremes 3 1. INTRODUCTION, LIMITING RESULTS IN THE FIELD OF EXTREMES AND PARAMETRIC APPROACHES We shall assume that we have a sample (X1, ...,Xn) of n independent, identically distributed (IID) or possibly stationary, weakly dependent random variables from an underlying cumulative distribution function (CDF), F , and shall use the notation (X X ) for the sample of associated ascending 1,n ≤ ··· ≤ n,n order statistics (OSs). Statistics of univariate extremes (SUE) helps us to learn from disastrous or almost disastrous events, of high relevance in society and with a high societal impact. The domains of application of SUE are thus quite diverse. We mention the fields of hydrology, meteorology, geology, insurance, finance, structural engineering, telecommunications and biostatistics (see, for instance, and among others, Coles, 2001; Reiss & Thomas, 2001, 2007; Beirlant et al., 2004, 1.3; Castillo et al., 2005; Resnick, 2007). Although it is possible to find § some historical papers with applications related to extreme events, the field dates back to Gumbel, in papers from 1935 on, summarized in his book (Gumbel, 1958). Gumbel develops statistical procedures essentially based on Gnedenko’s (Gnedenko, 1943) extremal types theorem (ETT), one of the main limiting results in the field of extreme value theory (EVT), briefly summarized below. 1.1. Main limiting results in EVT The main limiting results in EVT date back to the papers by Fr´echet (1927), Fisher & Tippett (1928), von Mises (1936) and Gnedenko (1943). Gnedenko’s ETT provides the possible limiting behaviour of the sequence of maximum or minimum values, linearly normalised, and an incomplete characterization, fully achieved in de Haan (1970), of the domains of attraction of the so-called max- stable (MS) or min-stable laws. Here, we shall always deal with the right-tail, F (x) := 1 F (x), for large x, i.e., we shall deal with top OSs. But all results − for maxima (top OSs) can be easily reformulated for minima (low OSs). Indeed, n X1,n = max1≤i≤n( Xi), and consequently, P(X1,n x) = 1 1 F (x) . MS − − ≤ n− { − } laws are defined as laws S such that the functional equation S (αn x+βn) = S(x), n 1, holds for some α > 0, β R. More specifically, all possible non-degener- ≥ n n ∈ ate weak limit distributions of the normalized partial maxima Xn,n, of IID random variables X1, ...,Xn, are (generalized) extreme value distributions (EVDs), i.e., if there are normalizing constants a > 0, b R, and some non-degenerate CDF G n n ∈ such that, for all x, (1.1) lim P (Xn,n bn)/an x = G(x) , n→∞ − ≤ 4 J. Beirlant, Frederico Caeiro and M. Ivette Gomes we can redefine the constants in such a way that exp (1 + γ x)−1/γ , 1 + γ x > 0, if γ = 0 , (1.2) G(x) Gγ(x) := − 6 ≡ ( exp exp( x) , x R, if γ = 0 , − − ∈ given here in the von Mises–Jenkinson form (von Mises, 1936; Jenkinson, 1955). If (1.1) holds, we then say that the CDF F which is underlying X1,X2, ..., is in the max-domain of attraction (MDA) of Gγ, in (1.2), and often use the notation F M(G ). The limiting CDFs G in (1.1) are then MS. They are indeed the ∈ D γ unique MS laws. The real parameter γ, the primary parameter of interest in extreme value analysis (EVA), is called the extreme value index (EVI). The EVI, γ, governs the behaviour of the right-tail of F . The EVD, in (1.2), is often separated into the three types: Type I (Gumbel) : Λ(x) = exp exp( x) , x R , − − ∈ (1.3) Type II (Fr´echet) : Φ (x) = exp( x−α), x 0 , α − ≥ Type III (max-Weibull) : Ψ (x) = exp ( x)α x 0 . α − − ≤ Indeed, with γ = 0, γ = 1/α > 0 and γ = 1/α < 0, respectively, we have Λ(x) = − G (x), Φ (x) = G α(1 x) and Ψ (x) = G α(x + 1) , with G the EVD 0 α 1/α{ − } α −1/α{ } γ in (1.2).
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