Generalized method of L-moment estimation for stationary and nonstationary extreme value models
Abstract
Precisely estimating out-of-sample upper quantiles is very important in risk assessment and in engineering practice for structural design to prevent a greater disaster. For this purpose, the generalized extreme value (GEV) distribution has been broadly used. To estimate the parameters of GEV distribution, the maximum likelihood estimation (MLE) and L-moment estimation (LME) methods have been primarily employed. For a better estimation using the MLE, several studies considered the generalized MLE (penalized likelihood or Bayesian) methods to cooperate with a penalty function or prior information for parameters. However, a generalized LME method for the same purpose has not been developed yet in the literature. We thus propose the generalized method of L-moment estimation (GLME) to cooperate with a penalty function or prior information. The proposed estimation is based on the generalized L-moment distance and a multivariate normal likelihood approximation. Because the L-moment estimator is more efficient and robust for small samples than the MLE, we reasonably expect the advantages of LME to continue to hold for GLME. The proposed method is applied to the stationary and nonstationary GEV models with two novel (data-adaptive) penalty functions to correct the bias of LME. A simulation study indicates that the biases of LME are considerably corrected by the GLME with slight increases in the standard error. Applications to US flood damage data and maximum rainfall at Phliu Agromet in Thailand illustrate the usefulness of the proposed method. This study may promote further work on penalized or Bayesian inferences based on L-moments.
Summary
This paper addresses the problem of accurately estimating upper quantiles of extreme value distributions, which is crucial for risk assessment and engineering design. The authors propose a novel "Generalized method of L-moment estimation" (GLME) to improve upon the traditional L-moment estimation (LME), especially when dealing with small sample sizes. LME is known to be more robust and efficient than Maximum Likelihood Estimation (MLE) in small samples, but lacks a straightforward way to incorporate prior information or penalty functions, unlike generalized MLE (GMLE). The GLME method bridges this gap by incorporating a penalty function or prior information through a generalized L-moment distance and a multivariate normal likelihood approximation. The GLME method is applied to both stationary and non-stationary Generalized Extreme Value (GEV) models. The authors introduce two novel, data-adaptive penalty functions designed to correct the bias of the LME, particularly for the shape parameter of the GEV distribution. A simulation study demonstrates that GLME effectively corrects the bias of LME while only slightly increasing the standard error. The usefulness of the proposed method is further illustrated through applications to US flood damage data and maximum rainfall data in Thailand. The paper concludes that GLME can promote further research on penalized or Bayesian inferences based on L-moments, especially when prior information is available.
Key Insights
- •Novel GLME Method: The paper introduces a new GLME method based on generalized L-moment distance and a multivariate normal likelihood approximation, enabling the incorporation of penalty functions or prior information into L-moment estimation.
- •Data-Adaptive Penalty Functions: Two novel data-adaptive penalty functions are proposed for bias correction in LME, specifically targeting the shape parameter of the GEV distribution. These penalty functions are designed to address the tendency of LME to underestimate high quantiles when the shape parameter is significantly negative.
- •Bias Correction with GLME: The simulation study demonstrates that GLME with a beta penalty function significantly reduces the bias of LME, particularly for negative shape parameters, while maintaining a comparable standard error.
- •Applicability to Non-Stationary Models: The GLME method is successfully extended to non-stationary GEV models, providing a robust alternative to MLE for analyzing extreme events in changing environments.
- •Performance Metrics: The simulation study uses Bias, Standard Error (SE), and Root Mean Squared Error (RMSE) to quantify the performance of GLME compared to LME and MLE.
- •Limited to Specific Penalty Functions: The study primarily focuses on two penalty functions (normal and beta), leaving room for exploring other potential penalty functions or prior distributions.
- •Computational Cost: The authors acknowledge that GLME is computationally more expensive than LME, which could be a limitation for very large datasets.
Practical Implications
- •Improved Risk Assessment: The GLME method can lead to more accurate estimation of extreme event quantiles (e.g., return levels), which is crucial for risk assessment in various fields, including hydrology, climate science, and finance.
- •Better Engineering Design: Engineers can use GLME to improve the design of infrastructure (e.g., dams, bridges) by more accurately estimating extreme loads and stresses, thereby increasing safety and resilience.
- •Beneficiaries: Hydrologists, climatologists, engineers, and risk managers who work with extreme value analysis can benefit from the GLME method.
- •Implementation: Practitioners can implement GLME using the provided R code available on GitHub, adapting the penalty functions and hyperparameters to specific applications and datasets.
- •Future Research: The paper opens up avenues for future research, including exploring different penalty functions, applying GLME to other extreme value models (e.g., generalized Pareto distribution), and investigating Bayesian model averaging techniques based on L-moments.