Estimating the tail index: another algorithmic method

The tail index is a determinant parameter within extreme value theory. Under a semiparametric approach, one has often to choose the number of the largest order statistics to include in estimates. This is a hard task since it is not possible to know for sure where the tail of data really begins. This crucial topic has been largely addressed in literature and several methods were developed. In this paper we analyze, through simulation, a heuristic method and compare it with two very popular methodologies. It will be seen that the new method can be a good alternative.Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the ”Fundação para a Ciência e a Tecnologia”, through the Project PEstOE/ MAT/UI0013/2014

Analysis of estimation methods for the extremal index

Many datasets present time-dependent variation and short-term clustering within extreme values. The extremal index is a primary measure to evaluate clustering of high values in a stationary sequence. Estimation procedures are based on the choice of a threshold and/or a declustering parameter or a block size. Here we revise several different methods and compare them through simulation. In particular, we will see that a recent declustering methodology may be useful for the popular runs estimator and for a new estimator that works under the validation of a local dependence condition. An application to real data is also presented.Fundação para a Ciência e Tecnologia (FCT)info:eu-repo/semantics/publishedVersio

A study of exponential-type tails applied to Birnbaum-Saunders models

Birnbaum-Saunders distributions have increasingly been used in environmental sciences applications. A major concern is the adjustment of extreme quantiles. Environmental data have often tails in the Gumbel domain which corresponds to a null tail index and does not allow us to distinguish the different tail weights that might exist between distributions within this domain. Exponential-tail distributions form an important subgroup with the peculiarity of including a parameter that specifies the “penultimate” tail behavior. In particular, we analyze the penultimate tail behavior of Birnbaum- Saunders distributions. We find examples with“heavier” tails than the classical one that can better accommodate environmental data highly concentrated on the right tail. This is illustrated with an application.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

Heuristic tools for the estimation of the extremal index: a comparison of methods

Clustering of exceedances of a critical level is a phenomenon that concerns risk managers in many areas. The extremal index θ measures the propensity of the large observations in a dataset to cluster. Thus the estimation of θ is an important issue recurrently addressed in literature. Besides a declustering parameter, inference also depends on a threshold. This choice is actually a crucial topic and is transversal to many other extremal parameters. In this paper we analyze a threshold-free heuristic procedure. We also make comparisons with other heuristic procedures already developed within the extremal index estimation. Our study is based on simulation. We illustrate with an application to environmental data.This research was financed by Portuguese Funds through FCT — Fundação para a Ciência e a Tecnologia, through the projects UID/MAT/00013/2013 and UID/MAT/00006/2013 and by the Research Center CEMAT through the Project UID/Multi/04621/2013.info:eu-repo/semantics/publishedVersio

Smoothness of time series: a new approach to estimation

The assessment of the risk of occurrence of extreme phenomena is inherently linked to the theory of extreme values. In the context of a time series, the analysis of its trajectory toward a greater or lesser smoothness, i.e. presenting a lesser or greater propensity for oscillations, respectively, constitutes another contribution in the assessment of the risk associated with extreme observations. For example, a financial market index with successive oscillations between high and low values shows investors a more unstable and uncertain behavior. In stationary time series, the upper tail smoothness coefficient is described by the tail dependence coefficient, a well-known concept first introduced by Sibuya. This work focuses on an inferential analysis of the upper tail smoothness coefficient, based on subsampling techniques for time series. In particular, we propose an estimator with reduced bias. We also analyze the estimation of confidence intervals through a block bootstrap methodology and a test procedure to prior detect the presence or absence of smoothness. An application to real data is also presented.The author is very grateful to the reviewers for their comments and suggestions which greatly improved this work. The research of the author as partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020

Measuring extremal clustering in time series

The propensity of data to cluster at extreme values is important for risk assessment. For example, heavy rain over time leads to catastrophic floods. The extremal index is a measure of Extreme Values Theory that allows measurement of the degree of high-value clustering in a time series. Inference about the extremal index requires a prior choice of values for tuning parameters, which impacts the efficiency of existing estimators. In this work, we propose an algorithm that avoids these constraints. Performance is evaluated based on simulations. We also illustrate with real data.The author was financed by Portuguese Funds through FCT - Fundação para a Ciência e a Tecnologia within the Projects UIDB/00013/2020 and UIDP/00013/2020 of Centre of Mathematics of the University of Minho

Dependência extremal: risco de contágio de valores extremos

O fenómeno da globalização, juntamente com um relaxamento na supervisão dos mercados financeiros, tornou-os mais vulneráveis e mais dependentes entre si. A ocorrência de grandes perdas em mercados fortes acaba por se reflectir ao nível das principais bolsas mundiais, e vice-versa. A necessidade de medir esta interdependência extremal conduziu ao aparecimento de diversos coeficientes no seio da teoria multivariada de valores extremos. Neste trabalho apresentam-se coeficientes para a dependência extremal entre dois vetores aleatórios, estendendo medidas existentes na literatura. A estimação ser´a também abordada e uma ilustração do conceito será feita com dados reais.This research was financed by FEDER Funds through “Programa Operacional Factores de Competitividade - COMPETE” and by Portuguese Funds through FCT - ”Fundação para a Ciência e a Tecnologia”, within the Project PEst-OE/MAT/UI0013/201