Box-Cox Transformations and Bias Reduction in Extreme Value Theory

The Box-Cox transformations are used to make the data more suitable for statistical analysis. We know from the literature that this transformation of the data can increase the rate of convergence of the tail of the distribution to the generalized extreme value distribution, and as a byproduct, the bias of the estimation procedure is reduced. The reduction of bias of the Hill estimator has been widely addressed in the literature of extreme value theory. Several techniques have been used to achieve such reduction of bias, either by removing the main component of the bias of the Hill estimator of the extreme value index (EVI) or by constructing new estimators based on generalized means or norms that generalize the Hill estimator. We are going to study the Box-Cox Hill estimator introduced by Teugels and Vanroelen, in 2004, proving the consistency and asymptotic normality of the estimator and addressing the choice and estimation of the power and shift parameters of the Box-Cox transformation for the EVI estimation. The performance of the estimators under study will be illustrated for finite samples through small-scale Monte Carlo simulation studies

Desafios em Estatística de Extremos

Cheias, fogos, furacões, secas e outros acontecimentos extremos têm fornecido uma razão para os desenvolvimentos recentes da teoria de valores extremos (EVT, do inglês, extreme value theory). A estatística de extremos é hoje em dia confrontada com muitos desafios, especialmente em tópicos relacionados com a modelação de risco e a eficiência e robustez das metodologias que nos permitem compreender a complexidade dos acontecimentos extremos nas mais diversas áreas. O compromisso entre robustez e extremos necessita pois de novos desenvolvimentos e de novas abordagens. Para além da estimação do índice de valores extremos, o parâmetro fundamental em EVT, consideraremos a estimação de quantis extremais e de períodos de retornos de níveis elevados.info:eu-repo/semantics/publishedVersio

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The extreme value Birnbaum-Saunders model in athletics

The Birnbaum-Saunders (BS) model is a life distribution that has recently been largely studied and applied. A random variable following the BS distribution can be defined through a simple transformation of a standard normal. The BS model can thus be generalized by switching the standard normal distribution of the basis random variable, allowing the construction of more general classes of models. Among those models, we mention the extreme value Birnbaum-Saunders (EVBS) models, recently introduced in the literature, and based on results from extreme value theory. A real application to athletics data will be used to illustrate the methodology and to provide the way this model and related models can link with traditional extreme value analysis methods.Este trabalho é financiado por Fundos FEDER através do Programa Operacional Factores de Competitividade - COMPETE e por Fundos Nacionais através da FCT - Fundação para a Ciência e a Tecnologia no âmbito do projecto PEst-C/MAT/UI0013/2011.Este trabalho é financiado por Fundos FEDER através do Programa Operacional Factores de Competitividade - COMPETE e por Fundos Nacionais através da FCT - Fundação para a Ciência e a Tecnologia no âmbito do projecto PEst-C/MAT/UI0013/2011

The extreme value Birnbaum-Saunders model, its moments and an application in biometry

The Birnbaum-Saunders (BS) model is a life distribution that has been largely studied and applied. Recently, a new version of the BS distribution based on extreme value theory has been introduced, which is named extreme value Birnbaum-Saunders (EVBS) distribution. In this article, we provide some further details on the EVBS models that can be useful as a supplement to the already existing results. We use these models to analyze real survival time data of patients treated with alkylating agents for multiple myeloma. This analysis allow us to show the adequacy of these new statistical distributions and identify them as models useful for medical practitioners in order to obtain a prediction of the survival times of these patients and evaluate changes in the dose of their treatment.Fundação para a Ciência e a Tecnologia (FCT) - Pluriannual Funding Program, PTDC/FEDER, PEst-OE/MAT/UI0006/2011, FCT/OE, POCI 201

The Use of Generalized Means in the Estimation of the Weibull Tail Coefficient

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation of the Weibull tail coefficient on the power mean-of-order-. Consistency and asymptotic normality of the estimators under study are put forward. Their performance for finite samples is illustrated through a Monte Carlo simulation. It is always possible to find a negative value of (contrarily to what happens with the mean-of-order- estimator for the extreme value index), such that, for adequate values of the threshold, there is a reduction in both bias and root mean square error

Non-regular Frameworks and the Mean-of-Order p Extreme Value Index Estimation

Most of the estimators of parameters of rare and large events, among which we dis- tinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory, are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarithm of the power mean of order p = 0 of a certain set of statistics. Only for heavy tails, i.e. a positive EVI, quite common in many areas of application, and trying to improve the performance of the classical Hill EVI-estimators, instead of the aforementioned geometric mean, we can more generally consider the power mean of order-p (MOp) and build associated MOp EVI-estimators. The normal asymptotic behaviour of MOp EVI-estimators has already been obtained for p < 1/(2ξ), with consistency achieved for p < 1/ξ , where ξ denotes the EVI. We shall now consider the non-regular case, p ≥ 1/(2ξ ), a situation in which either normal or non-normal sum- stable laws can be obtained, together with the possibility of an ‘almost degenerate’ EVI-estimation

Resampling methodologies and reliable tail estimation

Resampling methodologies, like the generalised jackknife and the bootstrap are importanttools for a reliable semi-parametric estimation of parameters of extreme or even rare events. Among these parameters we mention the extreme value index, the primary parameter in statistics of extremes, and the extremal index, a measure of clustering of extreme events. Most of the semi-parametric estimators of these parameters show the same type of behaviour: nice asymptotic properties, but a high variance for small k, the number of upper order statistics used in the estimation, a high bias for large k, and the need for an adequate choice of k. After a brief reference to some estimators of the aforementioned parameters and their asymptotic properties we present algorithms for an adaptive reliable estimation of the extreme value and extremal index

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info:eu-repo/semantics/publishedVersio