On the block maxima method in extreme value theory: PWM estimators (1310.3222v2)

Abstract: In extreme value theory, there are two fundamental approaches, both widely used: the block maxima (BM) method and the peaks-over-threshold (POT) method. Whereas much theoretical research has gone into the POT method, the BM method has not been studied thoroughly. The present paper aims at providing conditions under which the BM method can be justified. We also provide a theoretical comparative study of the methods, which is in general consistent with the vast literature on comparing the methods all based on simulated data and fully parametric models. The results indicate that the BM method is a rather efficient method under usual practical conditions. In this paper, we restrict attention to the i.i.d. case and focus on the probability weighted moment (PWM) estimators of Hosking, Wallis and Wood [Technometrics (1985) 27 251-261].

An Expert Overview of "On the Block Maxima Method in Extreme Value Theory: PWM Estimators"

The paper "On the Block Maxima Method in Extreme Value Theory: PWM Estimators" by Ana Ferreira and Laurens de Haan presents a rigorous investigation of the block maxima (BM) method as applied to extreme value theory (EVT), specifically focusing on probability weighted moment (PWM) estimators. The paper delineates the conditions under which the BM method can be theoretically justified and offers a comparative analysis against the peaks-over-threshold (POT) method, which has traditionally garnered more theoretical attention.

The authors primarily restrict their attention to independent and identically distributed (i.i.d.) cases, considering the PWM estimators introduced by Hosking, Wallis, and Wood. These estimators are particularly relevant in hydrological and climatological applications due to their computational simplicity and robustness to small sample sizes. Notably, the paper extends the theoretical understanding of these estimators under the BM framework.

Key Contributions

1. Theoretical Justification of the BM Method:
   • The authors provide a comprehensive justification for the BM method, which is often intuitively applied without rigorous theoretical backing. The paper elucidates a second-order expansion condition that accounts for potential biases arising from the assumption that block maxima strictly follow an extreme value distribution.
2. Asymptotic Properties of PWM Estimators:
   • Through detailed asymptotic analysis, the paper establishes the asymptotic normality of PWM estimators under the BM approach. This analysis makes use of Brownian bridge sequences to characterize the statistical properties of these estimators, particularly under conditions where both the block size and the number of blocks tend towards infinity.
3. Comparison with POT Method:
   • The paper conducts a theoretical comparison between the BM and POT methods, considering both asymptotic variance and bias. Under common conditions, it demonstrates that the asymptotic variances for BM are consistently lower compared to POT. Although BM exhibits higher bias, the mean square errors under optimal sample sizes are more favorable for BM across a substantial range of practical conditions.
4. Insights into Practical Applications:
   • The paper suggests that the BM method shows substantial efficacy under typical practical scenarios, particularly when sample sizes surpass 200 years of data or when the number of block observations aligns with historical data patterns. It also highlights the circumstances where the BM method might provide better accuracy than the POT approach, which relies on the selection of exceedance observations based on a predefined threshold.

Implications and Future Work

This paper's implications are significant for theoretical advancements in EVT and its practical applications, particularly in fields that rely on modeling extreme phenomena such as meteorology, hydrology, and environmental science. The results indicating lower asymptotic variances with the BM approach under practical circumstances may encourage a reevaluation of methodological preferences in these fields.

The paper also opens avenues for future research, suggesting extensions to non-i.i.d. cases and potential applicability to maximum likelihood estimators. Further exploration of these extensions could provide even broader application and robustness of the BM method in diverse data environments.

Conclusion

Ferreira and de Haan’s work stands as a substantial contribution to the body of knowledge on EVT, offering insightful theoretical and practical advancements in understanding and applying the BM method with PWM estimators. Future developments stemming from this research could enhance methodological frameworks for analyzing extreme events across various scientific disciplines.