Dora Prata Gomes

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.

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.

Adaptive estimation of a tail shape second order parameter: A computational comparative study

In Statistics of Extremes, the tail shape second order parameter is a relevant parameter whenever we want to improve the estimation of first order parameters. We shall consider two semi-parametric estimators of the shape second order parameter, parameterized with a tuning parameter. We provide a Monte Carlo comparative simulation study of several algorithms for the choice of such tuning parameter and for an adaptive estimation of the shape second order parameter.

A Log Probability Weighted Moment Estimator of Extreme Quantiles

In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.

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.

The human chromosomal fragile sites more often involved in constitutional deletions and duplications – A genetic and statistical assessment

Human chromosomal fragile sites (CFSs) are heritable loci or regions of the human chromosomes prone to exhibit gaps, breaks and rearrangements. Determining the frequency of deletions and duplications in CFSs may contribute to explain the occurrence of human disease due to those rearrangements. In this study we analyzed the frequency of deletions and duplications in each human CFS. Statistical methods, namely data display, descriptive statistics and linear regression analysis were applied to analyze this dataset. We found that FRA15C, FRA16A and FRAXB are the most frequently involved CFSs in deletions and duplications occurring in the human genome.

Statistical analysis of extreme river flows

Floods are recurrent events that can have a catastrophic impact. In this work we are interested in the analysis of a data set of gauged daily flows from the Whiteadder Water river, Scotland. Using statistic techniques based on extreme value theory, we estimate several extreme value parameters, including extreme quantiles and return periods of high levels.

Computer intensive methods for improving the extremal index estimation

Resampling methodologies have revealed recently as important tools in semi-parametric estimation of parameters in the field of extremes. Among a few parameters of interest, we are here interested in the extremal index, a measure of the degree of local dependence in the extremes of a stationary sequence. Most semi-parametric estimators of this parameter 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. Two extremal index estimators are here considered: a classical one and a reduced-bias generalized jackknife estimator. Bootstrap and jackknife methodologies are applied for obtaining the “best block size” for resampling and then constructing the bootstrap version of those estimators, that have led to more stable sample paths. A large simulation study was performed for illustrating the behavior of the resampling procedure proposed.

Reduced bias and threshold choice in the extremal index estimation through resampling techniques

In Extreme Value Analysis there are a few parameters of particular interest among which we refer to the extremal index, a measure of extreme events clustering. It is of great interest for initial dependent samples, the common situation in many practical situations. Most semi-parametric estimators of this parameter show the same 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. The Mean Square Error, a measure that encompasses bias and variance, usually shows a very sharp plot, needing an adequate choice of k. Using classical extremal index estimators considered in the literature, the emphasis is now given to derive reduced bias estimators with more stable paths, obtained through resampling techniques. An adaptive algorithm for estimating the level k for obtaining a reliable estimate of the extremal index is used. This algorithm has shown good results, but some improvements are s...