Statistics and Risk Management - 28/10/2015

Wednesday, 28 October 2015, 2:00 p.m.

Lecturer: M. Ivette Gomes, Centro de Estatística e Aplicações, Faculdade de Ciências Universidade de Lisboa, Portugal

Title: "Reliability of Large Coherent Systems and Penultimate Approximations in Extreme Value Theory"

Local: Room 1.3, Edifício VII

Faculdade de Ciências e Tecnologia, Quinta da Torre, Caparica

Abstract: The rate of convergence of the sequence of linearly normalized maxima or minima to the corresponding non-degenerate extreme value (EV) limiting distribution for maxima (EVDM) or for minima (EVDm) is a relevant problem in the field of extreme value theory. In 1928, Fisher and Tippett observed that, for normal underlying parents, if we approximate the distribution of the suitably linearly normalized sequence of maxima not by the so-called Gumbel limiting distribution, associated with an extreme value index (EVI) x = 0, but by an adequate sequence of other EV distributions with an EVI xn = o(1) < 0, the approximation can be asymptotically improved.  Such approximations are usually called penultimate approximations and have been theoretically studied from different perspectives. Recently, this same topic has been revisited in the field of reliability, where any coherent system can be represented as either a series-parallel (SP), a series structure with components connected in parallel, or a parallel-series (PS) system, a parallel structure with components connected in series. Its lifetime can thus be written as the minimum of maxima or the maximum of minima. For large-scale coherent systems it can be sensible to assume that the number of system components goes to infinity. Then, the possible non-degenerate EV laws, EVDM and EVDm, are eligible candidates for the finding of adequate lower and upper bounds for such system’s reliability. However, just as mentioned above, such non-degenerate limit laws are better approximated by an adequate penultimate distribution in most situations. It is thus sensible to assess both theoretically and through Monte-Carlo simulations the gain in accuracy when a penultimate approximation is used instead of the ultimate one. Moreover, researchers have essentially considered penultimate approximations in the class of EVDM or EVDm, but we can easily consider a much broader scope for that type of approximations, and such a type of models surely deserves a deeper consideration under statistical backgrounds. Penultimate models seem to be possible and interesting alternatives to the classical models but have never been deeply used in the literature.