# Peak-over-Threshold Estimators for Spectral Tail Processes: Random vs Deterministic Thresholds

## TL;DR

This paper compares estimators of spectral tail processes using deterministic versus random thresholds, showing they share the same asymptotic behavior and that random thresholds may perform better in finite samples.

## Contribution

It proves the asymptotic equivalence of estimators based on deterministic and random thresholds for spectral tail processes.

## Key findings

- Estimators with random thresholds have the same limit distribution as deterministic ones.
- Simulation shows random thresholds perform slightly better in finite samples.
- Theoretical results support practical use of random thresholds in extreme value analysis.

## Abstract

The extreme value dependence of regularly varying stationary time series can be described by the spectral tail process. Drees, Segers and Warchol [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.