M. Manuela Neves

Generalized Jackknife-Based Estimators for Univariate Extreme-Value Modeling

In this article, we revisit the importance of the generalized jackknife in the construction of reliable semi-parametric estimates of some parameters of extreme or even rare events. The generalized jackknife statistic is applied to a minimum-variance reduced-bias estimator of a positive extreme value index—a primary parameter in statistics of extremes. A couple of refinements are proposed and a simulation study shows that these are able to achieve a lower mean square error. A real data illustration is also provided.

Modeling Extreme Events: Sample Fraction Adaptive Choice in Parameter Estimation

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.

Averages of Hill estimators

Averaging Hills estimators leads to a reduction in the volatility of Hills plot. We deal with a generalization of the procedure proposed by Resnick and Stărică (1997), and, propose alternatives, assymptotically equivalent at the respective optimal levels, but with more interesting sample paths. Asymptotic normality is derived for intermediate levels where the asymptotic bias may be non-null. A simulation study completes the asymptotic results and shows the advantages of the proposed estimators in the problem of choosing the number of the top order statistics to be used in the estimation of the tail index.

Resampling Methodologies and the Estimation of Parameters of Rare Events

In extreme value theory the extremal index is a key parameter that enables a straightforward extension of the classic results for the independent case to stationary processes, measuring the degree of local dependence in the largest observations. Its estimation is important not only by itself but also because of its effect on the estimation of other parameters of extreme events. The estimators considered in the literature, despite of having good asymptotic properties, show a strong dependence on the high level un, presenting a high variance for high levels and a high bias when the level decreases. It has been seen that the bias is the dominant component of the mean squared error of the semiparametric estimators presented in the literature. Resampling techniques have been applied in situations where classical statistical procedures are difficult to use, but for a dependent setup, the resampling has to be done using blocks of observations. An adaptive resampling approach, based on block‐bootstrap and Jackkni...

Adaptive Choice and Resampling Techniques in Extremal Index Estimation

This work deals with the application of resampling techniques together with the adaptive choice of a ‘tuning’ parameter, the block size, \(b\), to be used in the bootstrap estimation of the extremal index, that is a key parameter in extreme value theory in a dependent setup. Its estimation has been considered by many authors but some challenges still remain. One of these is the choice of the number of upper order statistics to be considered in the semiparametric estimation. Block-bootstrap and Jackknife-After-Bootstrap are two computational procedures applied here for improving the behavior of the extremal index estimators through an adaptive choice of the block size for the resampling procedure. A few results of a simulation study will be presented.

Bootstrap and Other Resampling Methodologies in Statistics of Extremes

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.

Geostatistical Analysis in Extremes: An Overview

Classical statistics of extremes is very well developed in the univariate context for modeling and estimating parameters of rare events. Whenever rain, snow, storms, hurricanes, earthquakes, and so on, happen the analysis of extremes is of primordial importance. However such rare events often present a temporal aspect, a spatial aspect or both. Classical geostatistics, widely used for spatial data, is mostly based on multivariate normal distribution, inappropriate for modeling tail behavior. The analysis of spatial extreme data, an active research area, lies at the intersection of two statistical domains: extreme value theory and geostatistics. Some statistical tools are already available for the spatial modeling of extremes, including Bayesian hierarchical models, copulas and max-stable random fields. The purpose of this chapter is to present an overview of basic spatial analysis of extremes, in particular reviewing max-stable processes. A real case study of annual maxima of daily rainfall measurements in the North of Portugal is slightly discussed as well the main functions in R environment for doing such analysis.

Exploring R for modeling spatial extreme precipitation data

Natural hazards such as high rainfall and windstorms arise due to physical processes and are usually spatial in its nature. Classical geostatistics, mostly based on multivariate normal distributions, is inappropriate for modeling tail behavior. Several methods have been proposed for the spatial modeling of extremes, among which max-stable processes are perhaps the most well known. They form a natural class of processes extending extreme value theory when sample maxima are observed at each site of a spatial process. Jointly with the theoretical framework for modeling and characterizing measures of dependence of those processes, to deal with free and open-source software is of great value for practitioners. In this note, we illustrate how R can be used for modeling spatial extreme precipitation data.

Revisiting Resampling Methods in the Extremal Index Estimation: Improving Risk Assessment

Extreme value theory is an area of primordial importance for modelling extreme risks, allowing to estimate and predict beyond the range of data available. Among several parameters of interest, the extremal index is a crucial parameter in a dependent set-up, characterizing the degree of local dependence in the extremes of a stationary sequence. Its estimation has been addressed by several authors but some difficulties still remain. Resampling computer intensive methodologies have been recently considered in a reliable estimation of parameters of rare events. However classical bootstrap cannot be applied and block bootstrap procedures need to be considered. The block size for resampling strongly affects the estimates and needs to be properly chosen. Here, procedures for the choice of the block size for resampling are revisited and an improvement of the methods used in previous works for that choice is also considered. A simulation study will illustrate the performance of the aforementioned procedures. A real application is also presented.

Adaptive Choice of Thresholds and the Bootstrap Methodology: An Empirical Study

In this chapter, we discuss an algorithm for the adaptive estimation of a positive extreme value index, γ, the primary parameter in Statistics of Extremes. Apart from classical extreme value index estimators, we suggest the consideration of associated second-order corrected-bias estimators, and propose the use of bootstrap computer-intensive methods for the adaptive choice of thresholds.