On the Hill estimator: a comparison of methods

Extreme value theory (EVT) deals with the occurrence of extreme phenomena. The tail index is a very important parameter appearing in the estimation of the probability of rare events. Under a semiparametric framework, inference requires the choice of a number k of upper order statistics to be considered. This is the crux of the matter and there is no definite formula to do it, since a small k leads to high variance and large values of k tend to increase the bias. Several methodologies have emerged in literature, specially concerning the most popular Hill estimator (Hill, 1975). In this work we compare through simulation well-known procedures presented in Drees and Kaufmann (1998), Matthys and Beirlant (2000), Beirlant et al. (2002) and de Sousa and Michailidis (2004), with a heuristic scheme considered in Frahm et al. (2005) within the estimation of a different tail measure but with a similar context. We will see that the new method may be an interesting alternative.

Green's relations, regularity and abundancy for semigroups of quasi-onto transformations

Let V be an infinite-dimensional vector space and for every infinite cardinal n such that n≤dimV, let AE(V,n) denote the semigroup of all linear transformations of V whose defect is less than n. In 2009, Mendes-Gonçalves and Sullivan studied the ideal structure of AE(V,n). Here, we consider a similarly-defined semigroup AE(X,q) of transformations defined on an infinite set X. Quite surprisingly, the results obtained for sets differ substantially from the results obtained in the linear setting.

Hierarchical modelling of timber reference properties using probabilistic methods: Maximum Likelihood Method, Bayesian Methods and Probability Networks

In recent decades, an increased interest has been evidenced in the research on multi-scale hierarchical modelling in the field of mechanics, and also in the field of wood products and timber engineering. One of the main motivations for hierar-chical modelling is to understand how properties, composition and structure at lower scale levels may influence and be used to predict the material properties on a macroscopic and structural engineering scale. This chapter presents the applicability of statistic and probabilistic methods, such as the Maximum Likelihood method and Bayesian methods, in the representation of timber’s mechanical properties and its inference accounting to prior information obtained in different importance scales. These methods allow to analyse distinct timber’s reference properties, such as density, bending stiffness and strength, and hierarchically consider information obtained through different non, semi or destructive tests. The basis and fundaments of the methods are described and also recommendations and limitations are discussed. The methods may be used in several contexts, however require an expert’s knowledge to assess the correct statistic fitting and define the correlation arrangement between properties.