🤖 AI Summary
To address unreliable extreme value index (EVI) inference caused by scarcity of tail data, this paper proposes a perturbation-based synthetic exceedance generation method: controlled noise is injected into exceedances above a high threshold, followed by generalized Pareto distribution (GPD) modeling and construction of a consistent pivotal statistic. Innovatively, the perturbation mechanism is integrated with differential privacy guarantees; when GPD approximation error is substantial, a refined perturbation strategy is further introduced to enhance robustness. Experiments demonstrate that the proposed method significantly outperforms existing EVI inference approaches in terms of confidence interval coverage, width control, and resilience to model misspecification. It establishes a novel paradigm for reliable extreme-value analysis under sparse tail regimes.
📝 Abstract
The extreme value index (EVI) characterizes the tail behavior of a distribution and is crucial for extreme value theory. Inference on the EVI is challenging due to data scarcity in the tail region. We propose a novel method for constructing confidence intervals for the EVI using synthetic exceedances generated via perturbation. Rather than perturbing the entire sample, we add noise to exceedances above a high threshold and apply the generalized Pareto distribution (GPD) approximation. Confidence intervals are derived by simulating the distribution of pivotal statistics from the perturbed data. We show that the pivotal statistic is consistent, ensuring the proposed method provides consistent intervals for the EVI. Additionally, we demonstrate that the perturbed data is differentially private. When the GPD approximation is inadequate, we introduce a refined perturbation method. Simulation results show that our approach outperforms existing methods, providing robust and reliable inference.