On Estimation and Testing for Pareto Tails

2010 Mathematics Subject Classification: 62F10, 62F12.The t-Hill estimator for independent data was introduced by Fabian and Stehlik (2009). It estimates the extreme value index of distribution function with regularly varying tail. This paper considers sampling of an infinite moving average model. We prove that in the discussed case the t-Hill estimator is weak consistent. However, in contrast to independent identically distributed case here it is shown that the t-Hill and the Hill estimator applied to the moving average model are not robust with respect to large observations

Weak properties and robustness of t-Hill estimators

International audienceWe describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust 2 P. Jordanova et al. tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile

Structural and Mechanical Changes of AlMgSi0.5_{0.5} Alloy during Extrusion by ECAP Method

SPD (several plastic deformations) methods make it possible to obtain an ultrafine-grained structure (UFG) in larger volumes of material and thus improve its mechanical properties. The presented work focuses on the structural and mechanical changes of aluminium alloy AlMgSi$_{0.5}$ (EN AW 6060) during processing by repeated extrusion through the ECAP rectangular channel. After a four-pass extrusion, the samples’ microstructures were observed using an optical microscope, where refinement of the material grains was confirmed. Tensile tests determined the extrusion forces and allowed interpretation of the changes in the mechanical properties of the stressed alloy. The grain size was refined from 28.90 μm to 4.63 μm. A significant improvement in the strength of the material (by 45%) and a significant deterioration in ductility (to 60%) immediately after the first extrusion was confirmed. The third pass through the die appeared to be optimal for the chosen deformation path, while after the fourth pass, micro-cracks appeared, significantly reducing the strength of the material. Based on the measurement results, new analytical equations were formulated to predict the magnitude or intensity of the volumetric and shape deformations of the structural grain size and, in particular, the adequate increase in the strength and yield point of the material