Ordinal patterns in clusters of subsequent extremes of regularly varying time series
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Ordinal patterns in clusters of subsequent extremes of regularly varying time series Marco Oesting1,2
· Alexander Schnurr1
Received: 4 February 2019 / Revised: 20 July 2020 / Accepted: 4 August 2020 / © The Author(s) 2020
Abstract In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine. Keywords Cluster · Ordinal pattern · Peaks-over-threshold · Regularly varying time series · Tail process AMS 2000 Subject Classifications 62M10 · 62G32
1 Introduction In time series data sets, extremes often do not occur at scattered instants of time, but tend to form clusters. Assigning a cluster of extremes to a single extreme event, such as a flood in the context of a hydrological time series or a stock market crash in the context of a financial data, the distribution of these clusters is crucial for risk assessment.
Marco Oesting
[email protected]; [email protected] 1
Department Mathematik, University of Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
2
University of Stuttgart, Stuttgart Center for Simulation Science (SC SimTech) & Institute for Stochastics and Applications, Allmandring 5b, 70569 Stuttgart, Germany
M. Oesting, A. Schnurr
In order to analyze the occurrence times of extremes defined as exceedances over some high threshold u, some profound theory has been built up since the 1970s. Within this framework, data X1 , . . . , Xn from a stationary time series (Xt )t∈Z are typically divided into different blocks. Then, repeated extremes are said to form a cluster if they occur within the same temporal block. Due to the convergence of the process of exceedances to a Poisson point process under appropriate conditions as u → ∞, the distribution of these clusters converges weakly provided that the block size increases at the right speed. The limit distribution is nicely linked to the wellknown concept of the extremal index of the time series which can be interpreted as the reciprocal of the mean limiting cluster size (cf. Leadbetter et al. 1983; Embrechts and Kl¨uppelberg 1997; Chavez-Demoulin and Davison 2012, for an overview). Besides the extremal index, several other cluster characteristics are of interest and can be estimated, such as the distribution of the cluster size (Robert 2
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