保守尾指数估计的大偏差和中偏差

中文摘要 AI 辅助

为了设计能够抵御比观测记录中更罕见事件的系统，需要使用极值方法来外推分布尾部。尾指数估计器（如Hill估计器）是这种外推的核心，但高估尾指数可能导致对罕见事件概率的严重低估。受此启发，我们推导了Hill估计器的大偏差和中偏差渐近性质，并利用它们构造估计器，使其超过真实尾指数的概率以受控的指数速率（衰减速率）衰减。在大偏差范围内，我们证明，在基于相同顶部$k$个次序统计量的尺度不变估计器中，Hill估计器的一个简单重缩放版本在偏差和衰减速率之间实现了最优平衡。在二阶条件下，我们量化了Hill偏差的影响，分析了偏差校正估计器，并确定了在二阶参数$\ ho$等于零的边界情况下中偏差的充分条件。

英文摘要

To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top $k$ order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter $ρ$ equals zero.

发表机构

• University of Twente（特温特大学）
• CWI（荷兰数学与计算机科学中心）
• TU Eindhoven（埃因霍温理工大学）

机构由 AI 辅助整理，请以论文原文为准。