Variance-reduced extreme value index estimators using control variates in a semi-supervised setting

TL;DR

This work tackles the high variance problem in Extreme Value Index estimation caused by limited extreme observations by introducing a semi-supervised transfer-learning approach based on approximate control variates. The transferred Hill estimator achieves variance reduction that scales with the tail dependence between target and source data and remains largely independent of the actual EVI values, enabling robust gains even when tail heaviness differs. The methodology extends to the moment estimator and demonstrates practical variance reductions in multi-fidelity water surge and ice accretion applications, illustrating its broad applicability in engineering risk assessment. Overall, the framework provides a principled, bias-free variance reduction tool for EVI estimation and motivates extensions to other estimators and settings.

Abstract

The estimation of the Extreme Value Index (EVI) is fundamental in extreme value analysis but suffers from high variance due to reliance on only a few extreme observations. We propose a control variates based transfer learning approach in a semi-supervised framework, where a small set of coupled target and source observations is combined with abundant unpaired source data. By expressing the Hill estimator of the target EVI as a ratio of means, we apply approximate control variates to both numerator and denominator, with jointly optimized coefficients that guarantee variance reduction without introducing bias. We show theoretically and through simulations that the asymptotic relative variance reduction of the transferred Hill estimator is proportional to the tail dependence between the target and source variables and independent of their EVI values. Thus, substantial variance reduction can be achieved even without similarity in tail heaviness of the target and source distributions. The proposed approach can be extended to other EVI estimators expressed with ratio of means, as demonstrated on the moment estimator. The practical value of the proposed method is illustrated on multi-fidelity water surge and ice accretion datasets.

Figures (14)

• Figure 1: Boxplots of EVI estimations for different methods with strong target-source dependence ($\gamma^T=0.25$, $\gamma^S=0.5$, $n=1,000$, $k=100$, $m=5,000$, $\theta=10$)
• Figure 2:
• Figure 3: Variance reduction according to the target-source tail dependence ($\gamma^T=0.25$, $\gamma^S=0.5$, $n=1,000$, $k=100$, $m=5,000$)
• Figure 4: ($\gamma^T=0.25$, $\gamma^S=0.5$, $n=1,000$, $k=100$, $\theta=5$)
• Figure 5: ($\gamma^T=0.25$, $\gamma^S=0.5$, $m=5,000$, $\theta=5$)

Theorems & Definitions (12)

• Theorem 1: fisher_limiting_1928gnedenko_sur_1943
• Theorem 2: pickands_iii_statistical_1975balkema_residual_1974
• Definition 1: First-order extended regular variation de_haan_slow_1984
• Definition 2: Second-order extended regular variation alves_note_2007
• Definition 3: Hill estimator hill_simple_1975
• Definition 4: Moment estimator dekkers_moment_1989
• Definition 5: ACV/ACV estimator bocquet_control_2025
• Definition 6: Transferred Hill estimator
• Proposition 1
• Proposition 2