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

Nonparametric estimation of the tail-dependence coefficient

A common measure of tail dependence is the so-called tail-dependence coefficient. We present a nonparametric estimator of the tail-dependence coefficient and prove its strong consistency and asymptotic normality in the case of known marginal distribution functions. The finite-sample behavior as well as robustness will be assessed through simulation. Although it has a good performance, it is sensitive to the extreme value dependence assumption. We shall see that a block maxima procedure might improve the estimation. This will be illustrated through simulation. An application to financial data shall be presented at the end.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

Estimating the extremal index through the tail dependence concept

The extremal index θ is an important parameter in extreme value analysis when extending results from independent and identically distributed sequences to stationary ones. A connection between the extremal index and the tail dependence coefficient allows the introduction of new estimators. The proposed ones are easy to compute and we analyze their performance through a simulation study. Comparisons with other existing methods are also presented. Case studies within environment are considered in the end.info:eu-repo/semantics/acceptedVersio

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

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

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

Contributions for modeling characterization of heavy-tail time series

The occurrence of extreme phenomena and their devastating impact have been on the agenda, especially in areas of environmental and economic-financial sciences, extending to insurance activity. The theory of extreme values allows an adequate approach in the statistical study of data associated with this type of phenomena. Heavy tail models thus play an important role and are increasingly a resource. In this work we will revisit some max/min-autoregressive and maximum-moving models and contribute to their characterization by deriving their autocorrelation structure based on the Spearman and Kendall coefficients, both useful tools in the identification of models in real data applications.The author was financed by Portuguese Funds through FCT - Fundação para a Ciência e a Tecnologia within the Projects UID/MAT/00013/2013, UID/MAT/00006/2019 and by the research center CEMAT (Instituto Superior T´ecnico, Universidade de Lisboa) through the Project UID/Multi/04621/2013

Clustering of extreme values: estimation and application

The extreme value theory (EVT) encompasses a set of methods that allow inferring about the risk inherent to various phenomena in the scope of economic, financial, actuarial, environmental, hydrological, climatic sciences, as well as various areas of engineering. In many situations the clustering effect of high values may have an impact on the risk of occurrence of extreme phenomena. For example, extreme temperatures that last over time and result in drought situations, the permanence of intense rains leading to floods, stock markets in successive falls and consequent catastrophic losses. The extremal index is a measure of EVT associated with the degree of clustering of extreme values. In many situations, and under certain conditions, it corresponds to the arithmetic inverse of the average size of high-value clusters. The estimation of the extremal index generally entails two sources of uncertainty: the level at which high observations are considered and the identification of clusters. There are several contributions in the literature on the estimation of the extremal index, including methodologies to overcome the aforementioned sources of uncertainty. In this work we will revisit several existing estimators, apply automatic choice methods, both for the threshold and for the clustering parameter, and compare the performance of the methods. We will end with an application to meteorological data.The author is very grateful for the valuable comments of the reviewer and the associate editor that contributed to the improvement of the article. The research at CMAT was partially financed by Portuguese Funds through FCT (Fundacao para a Ciencia e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020