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

2025-11-19

AI Summary

Extreme value index (EVI) estimation suffers from high variance due to reliance on a small number of upper tail observations. Method: We propose a semi-supervised transfer learning framework grounded in the control variates method, leveraging a small set of paired data and abundant unpaired source-domain data to reduce estimation variance. Crucially, we integrate control variates into extreme-value estimation for the first time, jointly optimizing coefficients to achieve substantial asymptotic variance reduction while preserving unbiasedness. Our approach uses the ratio-mean of Hill and moment estimators as the baseline and embeds transfer learning architecture. Contribution/Results: The variance reduction depends solely on tail dependence—not on matching tail thickness between source and target domains. Theoretical analysis and simulations confirm significant asymptotic relative variance reduction. Empirical validation on multi-fidelity surge and ice-accumulation datasets demonstrates practical effectiveness.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

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.

Problem

• Reducing high variance in extreme value index estimation
• Using control variates with unpaired source data
• Achieving variance reduction without requiring similar tail distributions

Innovation

• Control variates reduce variance in EVI estimation
• Semi-supervised transfer learning with coupled observations
• Jointly optimized coefficients ensure unbiased variance reduction