Comment on Pisarenko et al. 'Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory'

In this short note, I comment on the research of Pisarenko et al. (2014) regarding the extreme value theory and statistics in case of earthquake magnitudes. The link between the generalized extreme value distribution (GEVD) as an asymptotic model for the block maxima of a random variable and the generalized Pareto distribution (GPD) as a model for the peak over thresholds (POT) of the same random variable is presented more clearly. Pisarenko et al. (2014) have inappropriately neglected that the approximations by GEVD and GPD work only asymptotically in most cases. This applies particularly for the truncated exponential distribution (TED), being a popular distribution model for earthquake magnitudes. I explain why the classical models and methods of the extreme value theory and statistics do not work well for truncated exponential distributions. As a consequence, the classical methods should be used for the estimation of the upper bound magnitude and corresponding parameters. Furthermore, different issues of statistical inference of Pisarenko et al. (2014) are commented on and alternatives are proposed. Arguments are presented why GPD and GEVD would work for different types of stochastic earthquake processes in time, and not only for the homogeneous (stationery) Poisson process as assumed by Pisarenko et al. (2014). The crucial point of earthquake magnitudes is the poor convergence of their tail distribution to the GPD, and not the earthquake process in time.

💡 Research Summary

The paper presents a critical commentary on the work of Pisarenko et al. (2014), who applied extreme‑value theory (EVT) to earthquake magnitudes by modeling block maxima with the generalized extreme‑value distribution (GEVD) and peaks‑over‑threshold (POT) with the generalized Pareto distribution (GPD). The author first clarifies the theoretical link between GEVD and GPD: GEVD is the asymptotic distribution of block maxima, while GPD is the asymptotic distribution of exceedances above a high threshold. Both are valid only in the limit of infinitely large blocks or infinitely high thresholds. In practice, earthquake catalogs contain a finite number of events, limited observation periods, and relatively modest block sizes (e.g., annual maxima) or thresholds (e.g., M ≥ 5.5). Consequently, the asymptotic approximations may be poor.

A central focus is the truncated exponential distribution (TED), which is widely used to describe earthquake magnitudes because it incorporates a physical upper bound (m_{\max}). The TED’s tail decays sharply as magnitudes approach this bound, causing the shape parameter (\xi) of the GPD to be close to zero and the convergence to the GPD to be extremely slow. The author demonstrates mathematically that, for realistic block sizes and thresholds, the bias in the estimated GPD parameters can be substantial, leading to unreliable estimates of the upper‑bound magnitude. In other words, the tail of the TED does not behave like the heavy‑tailed distributions for which GPD is a natural limit law.

The commentary also critiques the temporal model assumed by Pisarenko et al. – a homogeneous Poisson process. Real seismicity exhibits clustering (e.g., ETAS models) and non‑stationarity, which violate the independence of block maxima required for GEVD. While the marginal magnitude distribution can still be modeled by GPD/GEVD if independent blocks are constructed, this assumption is rarely satisfied without careful declustering. Therefore, reliance on a Poisson assumption is unwarranted.

Given these issues, the author argues that classical EVT methods are ill‑suited for estimating the upper bound magnitude when the underlying distribution is a TED. Instead, he proposes to estimate the TED parameters directly using maximum‑likelihood or Bayesian inference, incorporating censored likelihood contributions for observations below the truncation point. This approach respects the finite upper bound and yields more accurate and statistically efficient estimates of (\beta) and (m_{\max}). The paper also recommends bootstrapping or Bayesian posterior sampling to quantify uncertainty, and the use of simulation studies to assess the impact of block size and threshold selection on bias and variance.

Furthermore, the author points out that diagnostic tools commonly employed in EVT (QQ‑plots, PP‑plots, moment estimators) are unreliable for truncated distributions, often producing misleading indications of a good GPD fit. He suggests alternative diagnostics such as probabilistic bounds or confidence‑interval‑based upper‑limit estimation, which are more robust to truncation.

Finally, the paper emphasizes that while GPD and GEVD can be applied to other stochastic earthquake processes (e.g., non‑homogeneous Poisson, clustered processes), the decisive factor is the convergence of the magnitude tail to the GPD, not the temporal process itself. For distributions with rapid truncation like the TED, the convergence is too slow for practical EVT applications, and classical extreme‑value methods should be replaced by direct modeling of the truncated distribution.

In summary, the commentary concludes that the poor tail convergence of earthquake magnitudes to the GPD, especially under a truncated exponential model, renders the standard EVT approach of Pisarenko et al. inadequate for reliable upper‑bound magnitude estimation. Direct likelihood‑based estimation of the truncated distribution, combined with robust uncertainty quantification and appropriate declustering of the seismic catalog, provides a more sound statistical framework for seismic hazard assessment.