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

Modelling (and forecasting) extremes in time series: A naive approach

In Extreme Value Theory, we are essentially interested in the estimation of quantities related to extreme events. Whenever the focus is in large values, estimation is usually performed based on the largest k order statistics in the sample or on the excesses over a high level u. Here we are interested in modelling (and forecast- ing) extremes in time series. For modelling and forecasting classical time series, Boot.EXPOS is a computational procedure built in the R environment that has revealed to perform quite well in a large number of forecasting competitions. However, to deal with extreme values, a modification of that algorithm needs to be considered and is here under studyN/

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

A study of the jackknife method in the estimation of the extremal index

Clustering of high values occurs in many real situations and affects inference on extremal events. For stationary dependent sequences, under general local and asymptotic dependence conditions, the degree of clustering is measured through a parameter called the extremal index. The estimation of extreme events or parameters is usually based on a k number of top order statistics or on the exceedances of a high threshold u and is very sensitive to either of these choices. In particular, the bias increases with a growing k and a decreasing u. The use of the Jackknife methodology may help reduce bias. We analyse this method through a simulation study applied to several estimators of the extremal index. An application to real data sets illustrates the results

Extreme Value Index Estimators and Smoothing Alternatives: A Critical Review

Extreme-value theory and corresponding analysis is an issue extensively applied in many different fields. The central point of this theory is the estimation of a parameter γ, known as the extreme-value index. In this paper we review several extreme-value index estimators, ranging from the oldest ones to the most recent developments. Moreover, some smoothing and robustifying procedures of these estimators are presented.Extreme value index, Semi-parametric estimation, Smoothing modification

Measuring Spatial Extremal Dependence

The focus of this thesis is extremal dependence among spatial observations. In particular, this research extends the notion of the extremogram to the spatial process setting. Proposed by Davis and Mikosch (2009), the extremogram measures extremal dependence for a stationary time series. The versatility and flexibility of the concept made it well suited for many time series applications including from finance and environmental science. After defining the spatial extremogram, we investigate the asymptotic properties of the empirical estimator of the spatial extremogram. To this end, two sampling scenarios are considered: 1) observations are taken on the lattice and 2) observations are taken on a continuous region in a continuous space, in which the locations are points of a homogeneous Poisson point process. For both cases, we establish the central limit theorem for the empirical spatial extremogram under general mixing and dependence conditions. A high level overview is as follows. When observations are observed on a lattice, the asymptotic results generalize those obtained in Davis and Mikosch (2009). For non-lattice cases, we define a kernel estimator of the empirical spatial extremogram and establish the central limit theorem provided the bandwidth of the kernel gets smaller and the sampling region grows at proper speeds. We illustrate the performance of the empirical spatial extremogram using simulation examples, and then demonstrate the practical use of our results with a data set of rainfall in Florida and ground-level ozone data in the eastern United States. The second part of the thesis is devoted to bootstrapping and variance estimation with a view towards constructing asymptotically correct confidence intervals. Even though the empirical spatial extremogram is asymptotically normal, the limiting variance is intractable. We consider three approaches: for lattice data, we use the circular bootstrap adapted to spatial observations, jackknife variance estimation, and subsampling variance estimation. For data sampled according to a Poisson process, we use subsampling methods to estimate the variance of the empirical spatial extremogram. We establish the (conditional) asymptotic normality for the circular block bootstrap estimator for the spatial extremogram and show L2 consistency of the variance estimated by jackknife and subsampling. Then, we propose a portmanteau style test to check the existence of extremal dependences at multiple lags. The validity of confidence intervals produced from these approaches and a portmanteau style test are demonstrated through simulation examples. Finally, we illustrate this methodology to two data sets. The first is the amount of rainfall over a grid of locations in northern Florida. The second is ground-level ozone in the eastern United States, which are recorded on an irregularly spaced set of stations

Classe de Ciências

info:eu-repo/semantics/publishedVersio