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

An LpLp −quantile methodology for estimating extreme expectiles

Quantiles are a fundamental concept in extreme-value theory. They can be obtained from a minimization framework using an absolute error loss criterion. The companion notion of expectiles, based on squared rather than absolute error loss minimization, has recently been receiving substantial attention from the fields of actuarial science, finance and econometrics. Both of these notions can actually be embedded in a common framework of $Lp$-quantiles, whose extreme value properties have been explored very recently. However, and even though this generalized notion of quantiles has shown potential for the estimation of extreme quantiles and expectiles, it has so far not been used in the estimation of extreme value parameters of the underlying distribution of interest. In this paper, we work in a context of heavy tails, which is especially relevant to actuarial science, finance, econometrics and natural sciences, and we construct an estimator of the tail index of the underlying distribution based on extreme $Lp$-quantiles. We establish the asymptotic normality of such an estimator and in doing so, we extend very recent results on extreme expectile and $Lp$-quantile estimation. We provide a discussion of the choice of $p$ in practice, as well as a methodology for reducing the bias of our estimator. Its finite-sample performance is evaluated on simulated data and on a set of real hydrological data

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

Classe de Ciências

info:eu-repo/semantics/publishedVersio