TY - JOUR

T1 - Measures of serial extremal dependence and their estimation

AU - Davis, Richard A.

AU - Mikosch, Thomas Valentin

AU - Zhao, Yuwei

PY - 2013

Y1 - 2013

N2 - The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.Serial extremal dependence is typically characterized by clusters of exceedances of high thresholds in the series. We start by discussing the notion of extremal index of a univariate sequence, i.e. the reciprocal of the expected cluster size, which has attracted major attention in the extremal value literature. Then we continue by introducing the extremogram which is an asymptotic autocorrelation function for sequences of extremal events in a time series. In this context, we discuss regular variation of a time series. This notion has been useful for describing serial extremal dependence and heavy tails in a strictly stationary sequence. We briefly discuss the tail process coined by Basrak and Segers to describe the dependence structure of regularly varying sequences in a probabilistic way. Max-stable processes with Fréchet marginals are an important class of regularly varying sequences. Recently, this class has attracted attention for modeling and statistical purposes. We apply the extremogram to max-stable processes. Finally, we discuss estimation of the extremogram both in the time and frequency domains.

AB - The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.Serial extremal dependence is typically characterized by clusters of exceedances of high thresholds in the series. We start by discussing the notion of extremal index of a univariate sequence, i.e. the reciprocal of the expected cluster size, which has attracted major attention in the extremal value literature. Then we continue by introducing the extremogram which is an asymptotic autocorrelation function for sequences of extremal events in a time series. In this context, we discuss regular variation of a time series. This notion has been useful for describing serial extremal dependence and heavy tails in a strictly stationary sequence. We briefly discuss the tail process coined by Basrak and Segers to describe the dependence structure of regularly varying sequences in a probabilistic way. Max-stable processes with Fréchet marginals are an important class of regularly varying sequences. Recently, this class has attracted attention for modeling and statistical purposes. We apply the extremogram to max-stable processes. Finally, we discuss estimation of the extremogram both in the time and frequency domains.

U2 - 10.1016/j.spa.2013.03.014

DO - 10.1016/j.spa.2013.03.014

M3 - Journal article

VL - 123

SP - 2575

EP - 2602

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 7

ER -