Right Endpoint of a Distribution in Gumbel Domain of Attraction – Statistical Inference
Isabel Fraga Alves1, Laurens de Haan2, Cláudia Neves3
1Department of Statistics and Operations Research and Center of Statistics and Applications, Faculty Sciences University Lisbon, Lisbon, Portugal; 2University Tilburg, CEAUL, University of Lisbon and Erasmus University, Netherlands; 3Department of Mathematics, University Aveiro, Aveiro, Portugal

Extreme events are defined as extreme high (or low) values of whatever random characteristic we are interested in. These values play an important role because they may correspond to abnormal or dangerous operating conditions. Classical statistical inference techniques provide a good description of central behaviour, but not of extreme events. Extreme Value Theory (EVT) gives a probabilistic framework to model extreme events. EVT describes the fluctuations of the maximum of a random sample with parent distribution function F, Mn = max(X1, …, Xn). Let xF denote the (possibly infinite) right endpoint of F, ie, xF = sup{x: F(x) < 1}. Then Mn converges to xF, with probability 1, as n approaches ∞. In statistical analysis of rare events, the Generalized Extreme Value is a unified version of the only three possible limits for the distribution of Mn, provided suitable normalization in scale and location, for a large enough sample size n. This is supported by EVT, which relies on the fundamental Theorem of Gnedenko(1943) on max-domains of attraction, comprising Fréchet, Weibull and Gumbel max-domains: Fréchet domain of attraction refers to dfs with polynomial decaying tails; Weibull domain to dfs light-tailed with finite right endpoint and Gumbel domain is the intermediate case which refers to a great variety of dfs possessing an exponential tail, having or not a finite xF. In fact, EVT is a general framework: the “heavy tail” case as been extensively addressed in the literature, but EVT can also deal with “thin tail”, or even “no tail” (finite xF) cases. Less attention has yet been paid to the problem of assessing the presence of a distribution F with finite xF.

In the Gumbel max-domain setup, statistical inference for the finite right endpoint represents an important challenge for our knowledge into practical applications of real-world data sets in fields as environmetrics, climatology, or sports.

Keywords: Extreme value theory; Endpoint estimation; Gumbel domain of attraction; Semi-parametric approach to statistics of extremes

Biography: Isabel Fraga Alves is Associate Professor with Habilitation at Faculty of Science of University of Lisbon, PhD Thesis in “thin tail”, past-Coordinator of Center of Statistics and Applications (2006-2009), Elected Member of International Statistical Institute, Member of Bernoulli Society for Mathematical Statistics and Probability, Portuguese Statistical Society, Portuguese Mathematical Society, referee of - AISM (Ann.Instit. of Statist. Math.), Bernoulli, CSDA (Computational Statistics and Data Analysis), Estadística, European Series in Applied and Industrial Mathematics Probability and Statistics (ESAIM: P&S), Extremes, JMVA (J. Multiv. Analysis), JSPI (Journal of Statistical Planning and Inference), Metrika, REVSTAT-Statistical Journal, Statistics for Industry and Technology. Scientific publications mainly focused on “thin tail”, in several International Journals, in collaboration with Ivette Gomes, Laurens de Haan, Cláudia Neves, Paulo Araújo Santos, among others. List of Publications at http://docentes.deio.fc.ul.pt/fragaalves/publications.htm.