When modeling extreme events, there are a few primordial parameters, among which we refer to the extreme value index (EVI) and the extremal index (EI). Under a framework related to large values, the EVI measures the right tail weight of the underlying distribution and the EI characterizes the degree of local dependence in the extremes of a stationary sequence. Most of the semiparametric estimators of these parameters show the same type of behavior: nice asymptotic properties but a high variance for small values of k, the number of upper order statistics used in the estimation, and a high bias for large values of k. This brings a real need for the choice of k. Choosing some well-known estimators of those two parameters, we revisit the application of a heuristic algorithm for the adaptive choice of k. A simulation study illustrates the performance of the proposed algorithm.

Spatial statistics deals with statistical methods in which spatial locations play an explicit role in the analysis of data. Spatial data are often modeled as a realization of a Gaussian process or a function of a Gaussian process. However there are events, such as rain, snow, storms, hurricanes, earthquakes, where extremes are of the main interest. Here multivariate normal distributions are inappropriate for modeling tail behavior. The natural class of processes for dealing with extremes is the max-stable processes. This work briefly reviews the approaches to the statistical modeling of spatial extremes. Models for max-stable processes will be considered in an application to the annual maxima of daily precipitation over the North of Portugal

The main objective of Statistics of Extremes is the estimation of probabilities of rare events. When extending the analysis of the limiting behaviour of the extreme values from independent and identically distributed sequences to stationary sequences a key parameter appears, the extremal index θ, whose accurate estimation is not easy. Here we focus on the estimation of θ using blocks estimators, that can be constructed by using disjoint or sliding blocks. The asymptotic properties for both procedures were studied and compared but both blocks estimators require the choice of a threshold and a block length. Some criteria have appeared for the choice of those nuisance quantities but some research is still needed. We will show how the threshold and the block size choices can affect the estimates. However the main objective of this work is to revisit another estimation procedure that only depends on the block length, although some conditions on the underlying process need to be verified. The associated estimator presents nice asymptotic properties, and for finite samples is here illustrated a stability criterion for choosing the block length and then obtaining the θ estimate. A large simulation study has been performed and an application to daily mean flow discharge rate in the hydrometric station of Fragas da Torre in river Paiva, data collected from 1 October 1946 to 30 April 2012 is done.

In Statistics of Extremes, the estimation of parameters of extreme or even rare events is usually done under a semi-parametric framework. The estimators are based on the largest k-ordered statistics in the sample or on the excesses over a high level u. Although showing good asymptotic properties, most of those estimators present a strong dependence on k or u with high bias when the k increases or the level u decreases. The use of resampling methodologies has revealed to be promising in the reduction of the bias and in the choice of k or u. Different approaches for resampling need to be considered depending on whether we are in an independent or in a dependent setup. A great amount of investigation has been performed for the independent situation. The main objective of this article is to use bootstrap and jackknife methods in the context of dependence to obtain more stable estimators of a parameter that appears characterizing the degree of local dependence on extremes, the so-called extremal index. A simulation study illustrates the application of those methods.