Veröffentlichungen

@phdthesis{68f18a11e2264279b51f43145c0abb40,

title = "Estimators for temporal dependence of extremes",

abstract = "For the understanding of the behavior of the extremes of a stationary time series, the analysis of the extremal dependence in time is of high importance. For quantities describing this temporal dependence of extreme events, block estimators are often used. Block estimators are defined as the average of statistics depending on blocks of standardized extreme observations. This blocks estimators can be defined by using so-called sliding blocks, or it can be defined as an average over disjoint blocks.The asymptotic analysis for disjoint blocks estimators can be performed using the central limit theorems of Drees and Rootz{\'e}n (2010). In this thesis, a generalized functional limit theorem for suitable empirical processes is derived. As a special case, for the first time this allows a systematic asymptotic analysis of sliding blocks estimators. Specifically, the asymptotic normality of the standardized sliding blocks estimator is proved under weak conditions. In general, both the sliding and the disjoint blocks estimator can be used for the same estimation problem. In this thesis, we prove that the sliding blocks estimator in the POT setting never has a larger asymptotic variance than the disjoint blocks estimator.Among the indexes describing specific aspects of the extremal dependence of time series are the so-called cluster indexes. In this thesis, we consider two cluster indexes: the well known extremal index and the newer stop-loss index. For both indexes, the asymptotic distributions of the estimation errors are derived on the basis of the general theory mentioned above and, for the family of stop-loss indexes, even process convergence is shown. In each case, we consider a sliding blocks estimator, the associated disjoint blocks estimator and a runs estimator. With the unified framework used in this thesis, it is shown that all three estimators for the extremal index have the same asymptotic distribution - a fact that was not yet known in the literature. The asymptotic result for the sliding blocks estimator is shown for the first time in this work. Under the assumption of regular variation, the spectral tail process describes the entire extremal dependence structure of a stationary time series. Thus, for the initial problem of describing the temporal dependence of extremes, the estimation of its distribution is of particular interest. In this thesis, a new type of estimator is proposed, which is based on an invariance principle of the distribution of the spectral tail process. This invariance principle can be used for the construction of estimators by means of a projection method. For the corresponding estimator of the probability that the spectral tail process at some fixed time point lies in a specific Borel set, the asymptotic normality is derived using the general results for sliding blocks estimators mentioned above. Asymptotic normality is proved for both a known and an estimated index of regular variation. The conditions required for these asymptotic results are all shown to be satisfied by the general example of solutions to stochastic recurrence equations. Simulation results show that this new projection based estimator mostly has smaller variance than estimators known from the literature. Moreover, this estimator also has the most stable performance in terms of the RMSE. Overall, the new estimator has some desirable properties that its predecessors from the literature do not possess.",

keywords = "extreme value theory, time series, nonparametric statistic, extremal dependency, empirical processes, uniform central limit theorem, Mathematik, Extremwertstatistik, Zeitreihenanalyse, Zentraler Grenzwertsatz, Nichtparametrische Statistik",

author = "Neblung, {Sebastian A.}",

year = "2021",

language = "English",

school = "University of Hamburg",