Robust and efficient estimation for the Generalized Extreme-Value distribution with application to flood frequency analysis in the UK

TL;DR

The paper develops a robust estimation framework for the three-parameter Generalized Extreme-Value (GEV) distribution using the Minimum Density Power Divergence (MDPD). By minimizing the H_α criterion, the authors achieve estimators that interpolate between the efficient maximum likelihood estimator (α=0) and robust L^2-type methods (α>0), and prove consistency, asymptotic normality, and bounded influence for α>0. Simulation studies show that MDPD maintains competitive efficiency in uncontaminated settings while delivering superior robustness under contamination, with the trade-off controlled by α. A UK flood-frequency case study demonstrates the method’s practical advantage by downweighting influential low floods without discarding data, yielding distributional fits and tail inferences closer to PILF-removed references. Overall, the work offers a theoretically sound and practically useful alternative for extreme-value analysis in environmental settings with contaminated data.

Abstract

A common approach for modeling extremes, such as peak flow or high temperatures, is the three-parameter Generalized Extreme-Value distribution. This is typically fit to extreme observations, here defined as maxima over disjoint blocks. This results in limited sample sizes and consequently, the use of classic estimators, such as the maximum likelihood estimator, may be inappropriate, as they are highly sensitive to outliers. To address these limitations, we propose a novel robust estimator based on the minimization of the density power divergence, controlled by a tuning parameter $α$ that balances robustness and efficiency. When $α= 0$, our estimator coincides with the maximum likelihood estimator; when $α= 1$, it corresponds to the $L^2$ estimator, known for its robustness. We establish convenient theoretical properties of the proposed estimator, including its asymptotic normality and the boundedness of its influence function for $α> 0$. The practical efficiency of the method is demonstrated through empirical comparisons with the maximum likelihood estimator and other robust alternatives. Finally, we illustrate its relevance in a case study on flood frequency analysis in the UK and provide some general conclusions in Section 6.

Figures (20)

• Figure 1: Asymptotic variance of the MDPD estimator involved in Theorem \ref{['th:asympt_norm']} for different values of $\alpha$ and of the ML estimator ($\alpha =0$) as a function of the shape parameter $\xi_0$, for the location parameter (left), the scale parameter (middle), and the shape parameter (right). The true parameter values are $\mu_0 = 0$ and $\sigma_0 = 1$.
• Figure 2: Influence functions of the MDPD estimators \ref{['eq:compo_IF']} for different values of $\alpha$ and of the ML estimator ($\alpha = 0$), for the location parameter (left), the scale parameter (middle), and the shape parameter (right). The true parameter values are $\mu_0 = 0$, $\sigma_0 = 1$, and $\xi_0 = -0.3$ corresponding to a upper bound for the domain of approximately 3.3. The x-axis represents the quantile level at which the influence functions are evaluated.
• Figure 3: Influence function of the MDPD estimators \ref{['eq:compo_IF']} for different values of $\alpha$ and of the ML estimator ($\alpha = 0$), for the location parameter (left), the scale parameter (middle), and the shape parameter (right). The true parameter values are $\mu_0 = 0$, $\sigma_0 = 1$, and $\xi_0 = 0.3$ corresponding to a lower bound for the domain of approximately -3.3. The x-axis represents the quantile level at which the influence functions are evaluated.
• Figure 4: Average Wasserstein distance over 200 replications (with standard errors) across various contaminated models. In the left panel, the shape parameter $\xi_1$ varies while the location and the scale parameters are fixed to the true model parameters ($\mu_1 = \mu_0$ and $\sigma_1=\sigma_0$). In the right panel, the scale parameter $\sigma_1$ varies while the location and the shape parameters are fixed to the true model parameters ($\mu_1 = \mu_0$ and $\xi_1=\xi_0$). Each sample has size $n = 100$, with contamination proportion $\varepsilon = 0.1$. The true model parameters are $\mu_0 = 0$, $\sigma_0 = 1$, and $\xi_0 = 0.1$.
• Figure 5: Average Wasserstein distance over 200 replications (with standard errors) across various contaminated models. In the left panel, the shape parameter $\xi_1$ varies while the location and the scale parameters are fixed to the true model parameters ($\mu_1 = \mu_0$ and $\sigma_1=\sigma_0$). In the right panel, the scale parameter $\sigma_1$ varies while the location and the shape parameters are fixed to the true model parameters ($\mu_1 = \mu_0$ and $\xi_1=\xi_0$). Each sample has size $n = 100$, with contamination proportion $\varepsilon = 0.1$. The true model parameters are $\mu_0 = 0$, $\sigma_0 = 1$, and $\xi_0 = 0$.

Theorems & Definitions (5)

• Theorem 3.1: Consistency and asymptotic normality
• Remark 3.1: Model misspecification.
• Proposition 3.1: Influence function
• Remark A.1: Minor Contribution
• Corollary B.1: Corollary A.3.4 in phdjuarez