Peak-over-threshold estimators for spectral tail processes: random vs deterministic thresholds

The extreme value dependence of regularly varying stationary time series can be described by the spectral tail process. Drees et al. (Extremes 18(3), 369-402, 2015) proposed estimators of the marginal distributions of this process based on exceedances over high deterministic thresholds and analyzed their asymptotic behavior. In practice, however, versions of the estimators are applied which use exceedances over random thresholds like intermediate order statistics. We prove that these modified estimators have the same limit distributions. This finding is corroborated in a simulation study, but the version using order statistics performs a bit better for finite samples.