ALBERT

Beschreibung:    The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the extremal dependence measure (EDM) and the extremogram . We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions. Content Type Journal Article Pages 1-26 DOI 10.1007/s10687-011-0135-9 Authors Martin Larsson, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA Sidney I. Resnick, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    In this paper, a new kind of location invariant Weiss-Hill estimator of the extreme value index γ  ∈ ℝ is proposed. The new estimator is a combination of two estimators proposed by Weiss (Nav Res Logist Q 1:111–114, 1971 ) and Fraga Alves (Extremes 4:199–217, 2001a ). The following properties of the new estimator are derived: weak consistency, strong consistency, and asymptotic expansions. A bias corrected version of the proposed estimator is given after determining an optimal sample fraction. Some finite simulation studies are performed. Content Type Journal Article Pages 1-34 DOI 10.1007/s10687-011-0134-x Authors Chengxiu Ling, Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Bâtiment Extranef, UNIL-Dorigny, 1015 Lausanne, Switzerland Zuoxiang Peng, School of Mathematics and Statistics, Southwest University, Chongqing, 400715 China Saralees Nadarajah, School of Mathematics, University of Manchester, Manchester, UK Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    Maxima of moving maxima of continuous functions (CM3) are max-stable processes aimed at modelling extremes of continuous phenomena over time. They are defined as Smith and Weissman’s M4 processes with continuous functions rather than vectors. After standardization of the margins of the observed process into unit-Fréchet, CM3 processes can model the remaining spatio-temporal dependence structure. CM3 processes have the property of joint regular variation. The spectral processes from this class admit particularly simple expressions given here. Furthermore, depending on the speed with which the parameter functions tend toward zero, CM3 processes fulfill the finite-cluster condition and the strong mixing condition. Processes enjoying these three properties also enjoy a simple expression for their extremal index. Next a method to fit CM3 processes to data is investigated. The first step is to estimate the length of the temporal dependence. Then, by selecting a suitable number of blocks of extremes of this length, clustering algorithms are used to estimate the total number of different profiles. The parameter functions themselves are estimated thanks to the output of the partitioning algorithms. The full procedure only requires one parameter which is the range of variation allowed among the different profiles. The dissimilarity between the original CM3 and the estimated version is evaluated by means of the Hausdorff distance between the graphs of the parameter functions. Content Type Journal Article Pages 1-31 DOI 10.1007/s10687-011-0136-8 Authors Thomas Meinguet, Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays 20, 1348 Louvain-la-Neuve, Belgium Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    There is a prevailing assumption that the largest inclusions in steel volumes follows mode I of the Generalized Extreme Values (GEV) distribution. In this work, the GEV distributions of non-metallic inclusions in six different high performance steels, of different grades and processing routes, were investigated by means of fractography of inclusions causing failure in ultrasonic fatigue testing to one billion cycles and all three modes of the GEV were found for the different steel grades. Values of the shape parameter ξ of the GEV distribution as high as 0.51 (standard deviation 0.11) were found in one steel grade. Thus, the present results show that the assumption of GEV-I (Gumbel, LEVD) distribution has to be substantiated before being used to estimate the size of the largest inclusions. Content Type Journal Article Pages 1-9 DOI 10.1007/s10687-011-0139-5 Authors Jens Ekengren, Department of Mechanical and Materials Engineering, Karlstad University, 651 88 Karlstad, Sweden Jens Bergström, Department of Mechanical and Materials Engineering, Karlstad University, 651 88 Karlstad, Sweden Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    Let ( X , Y ) = ( RU 1 , RU 2 ) be a given bivariate scale mixture random vector, with R  〉 0 independent of the bivariate random vector ( U 1 , U 2 ). In this paper we derive exact asymptotic expansions of the joint survivor probability of ( X , Y ) assuming that R has distribution function in the Gumbel max-domain of attraction, and ( U 1 , U 2 ) has a specific local asymptotic behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions. Content Type Journal Article Pages 1-20 DOI 10.1007/s10687-011-0129-7 Authors Enkelejd Hashorva, Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    We consider the extremal shot noise defined by M ( y )= sup { mh ( y - x );( x , m ) Î F }, where Φ is a Poisson point process on ℝ d × (0, + ∞ ) with intensity λdxG ( dm ) and h : ℝ d → [0, + ∞ ] is a measurable function. Extremal shot noises naturally appear in extreme value theory as a model for spatial extremes and serve as basic models for annual maxima of rainfall or for coverage field in telecommunications. In this work, we examine their properties such as boundedness, regularity and ergodicity. Connections with max-stable random fields are established: we prove a limit theorem when the distribution G is heavy-tailed and the intensity of points λ goes to infinity. We use a point process approach strongly connected to the Peak Over Threshold method used in extreme value theory. Properties of the limit max-stable random fields are also investigated. Content Type Journal Article Pages 1-30 DOI 10.1007/s10687-011-0131-0 Authors Clément Dombry, Laboratoire LMA, Université de Poitiers, Téléport 2, BP 30179, 86962 Futuroscope-Chasseneuil cedex, France Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung: Editorial: Special issue on marine safety and Rice’s formula Content Type Journal Article Pages 1-3 DOI 10.1007/s10687-011-0130-1 Authors Holger Rootzén, Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Gothenburg, Sweden Ross Leadbetter, Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    We consider a stationary and isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with value 0 or 1 according to whether a regular stationary process X is less or greater than a given level. The intervals corresponding to the points at which X is in a given phase are named chords. We are interested in obtaining information on the chord-length distribution functions. Working with the Palm probability measure and using level crossings techniques, in particular Rice methods, we can obtain not only the exact analytical formula of the chord-length distribution function but also the joint distribution function of the lengths of two successive chords. Finally, we indicate some concrete applications for the computation of usual stereological parameters. Content Type Journal Article Pages 1-20 DOI 10.1007/s10687-011-0141-y Authors Anne Estrade, MAP5 (UMR CNRS 8145), Université Paris Descartes, 45 rue des Saints-Pères, 75270 Paris 06, France Ileana Iribarren, Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela Marie Kratz, MAP5 (UMR CNRS 8145), ESSEC Business School Paris, Avenue Bernard Hirsch BP 50105, 95021 Cergy Pontoise Cedex, France Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    Let X 1 , X 2 , ⋯ be a sequence of independent and identically distributed random variables and M n  =  max { X 1 , X 2 , ⋯ , X n }. Suppose that some of the random variables X 1 , X 2 , ⋯ , X n can be observed and denote by the maximum of the observed random variables from the set { X 1 , X 2 , ⋯ , X n }. The limiting distribution of random vector is derived. The result is also extended to the case of stationary Gaussian sequences. In the end, the almost sure limit theorem on for a sequence of independent and identically distributed random variables is proved. Content Type Journal Article Pages 1-14 DOI 10.1007/s10687-011-0140-z Authors Zhongquan Tan, School of Mathematical Sciences, Soochow University, Suzhou, 215006 People’s Republic of China Yuebao Wang, School of Mathematical Sciences, Soochow University, Suzhou, 215006 People’s Republic of China Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Beschreibung:    The strong and the weak tail dependence coefficients are measures that quantify the probability of conjoint extreme events of two random variables. Whereas formulas for both tail dependence coefficients exist for the Gaussian and Student t distribution, only the strong tail dependence coefficient is known for their super-model, the elliptical generalized hyperbolic distribution, which is extremely popular in finance (see Schmidt 2003 ). In this work we derive a simple expression for the corresponding weak tail dependence coefficient using the mixture representation of the elliptical generalized hyperbolic distribution. Content Type Journal Article Pages 1-16 DOI 10.1007/s10687-011-0132-z Authors Stephan Schlueter, Department of Statistics and Econometrics, University of Erlangen-Nuremberg, Lange Gasse 20, 90403 Nuremberg, Germany Matthias Fischer, Department of Statistics and Econometrics, University of Erlangen-Nuremberg, Lange Gasse 20, 90403 Nuremberg, Germany Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999