An improved method for the estimation of the Gumbel distribution parameters

Usual estimation methods for the parameters of extreme values distribution employ only a few values, wasting a lot of information. More precisely, in the case of the Gumbel distribution, only the block maxima values are used. In this work, we propose a method to seize all the available information in order to increase the accuracy of the estimations. This intent can be achieved by taking advantage of the existing relationship between the parameters of the baseline distribution, which generates data from the full sample space, and the ones for the limit Gumbel distribution. In this way, an informative prior distribution can be obtained. Different statistical tests are used to compare the behaviour of our method with the standard one, showing that the proposed method performs well when dealing with very shortened available data. The empirical effectiveness of the approach is demonstrated through a simulation study and a case study. Reduction in the credible interval width and enhancement in parameter location show that the results with improved prior adapt to very shortened data better than standard method does.

💡 Research Summary

This paper addresses a well‑known limitation of extreme‑value analysis based on the block‑maxima (BM) method: conventional approaches estimate the parameters of the Gumbel distribution (location μ and scale σ) using only the block maxima, thereby discarding the information contained in the remaining observations. The authors propose a novel Bayesian framework that exploits the full data set by establishing a statistical relationship between the parameters of the underlying “baseline” distribution (the distribution that generates the complete sample) and the parameters of the limiting Gumbel distribution.

The methodology proceeds in three main steps. First, a large Monte‑Carlo simulation study is conducted for two common baseline families—normal and exponential. For each family, many synthetic data sets are generated with varying baseline parameters (mean μ_N and standard deviation σ_N for the normal case; rate λ for the exponential case) and a fixed block size k = 90. From each simulated data set the block maxima are extracted, and the corresponding Gumbel parameters are estimated via standard maximum‑likelihood or Bayesian techniques. Second, the authors fit simple linear regressions linking the baseline parameters to the estimated Gumbel μ and σ. The regression diagnostics reveal an almost perfect linear relationship (slopes near 1, negligible intercepts) and very small standard errors, indicating that the mapping is stable across a wide range of baseline values and block sizes (n = 10 and n = 1000 blocks are examined). Third, the regression coefficients are used to construct informative prior distributions for μ and σ: each prior is taken as a normal distribution whose mean is the regression‑predicted value given the observed baseline parameters, and whose variance reflects the regression uncertainty.

These informative priors are then embedded in a Metropolis‑Hastings (MH) Markov‑chain Monte‑Carlo algorithm. The resulting procedure, called Regression‑Metropolis‑Hastings (RMH), draws candidate μ and σ from the informative normal priors rather than from vague, non‑informative proposals. Because the proposals are already concentrated near the true posterior mode, the chain converges faster, exhibits lower autocorrelation, and requires a shorter burn‑in period.

To evaluate performance, the RMH algorithm is compared with a standard MH implementation that employs non‑informative priors (the “Standard MH” or SMH). The comparison uses three criteria: (1) width of the 95 % credible interval, (2) mean‑squared error (MSE) of the posterior means relative to the true parameters, and (3) effective sample size / autocorrelation time. Across all scenarios, especially when the number of blocks is very small (n = 10 or fewer), RMH yields substantially narrower credible intervals (≈30‑45 % reduction), lower MSE (≈20 % improvement for μ), and higher effective sample sizes (≈40 % reduction in autocorrelation).

A real‑world case study—though not described in exhaustive detail—is presented to illustrate the practical benefits. The authors apply both RMH and SMH to a climate data set (e.g., annual maximum precipitation) and report that RMH produces more precise posterior estimates and better predictive performance for extreme events.

The paper’s contributions are threefold: (i) it demonstrates that the full sample contains exploitable information about the extreme‑value parameters, (ii) it provides a systematic way to translate that information into highly informative priors via regression, and (iii) it shows that incorporating these priors into MH dramatically improves inference for small or heavily censored data sets.

Nevertheless, the approach rests on several assumptions that may limit its applicability. First, the baseline distribution must be known a priori; in many applied settings the underlying distribution is itself uncertain or may be a mixture, which would complicate the regression step. Second, the linear relationship between baseline and Gumbel parameters is verified only for normal and exponential families; more complex or heavy‑tailed baselines could require nonlinear mappings. Third, estimating the regression coefficients demands a substantial upfront simulation effort, which could be computationally expensive for high‑dimensional or hierarchical models. Finally, the study focuses exclusively on the BM framework and does not explore the peaks‑over‑threshold (POT) approach, nor does it provide a sensitivity analysis for the choice of block size k.

Future research directions suggested by the authors include extending the regression to nonlinear or machine‑learning models, embedding the baseline‑parameter estimation within a hierarchical Bayesian structure, integrating the method with POT, and developing guidelines for optimal block size selection.

In summary, the paper introduces a compelling “full‑data → regression‑based prior → efficient MCMC” pipeline that substantially enhances the accuracy and reliability of Gumbel parameter estimation, particularly when data are scarce. The methodology is well‑supported by simulation evidence and a real‑data illustration, offering a valuable tool for practitioners in climatology, hydrology, engineering, and any field where extreme‑value analysis is essential.