Weibull尾部系数估计的改进——一项比较研究
Improvements in the estimation of the Weibull tail coefficient -- a comparative study
AI总结：
本研究提出基于对数超出量幂的Weibull尾部系数新估计量，在二阶框架下减少偏差，并通过大规模蒙特卡洛模拟与现有估计量比较，验证其性能优势。
摘要：
Weibull尾部系数（WTC）在极值统计中处理Weibull型尾部时起着至关重要的作用。正态分布、Gamma分布、Weibull分布和Logistic分布等多种分布均表现出这种类型的尾部行为。WTC记为$θ$，是右尾函数$\bar F(x) :=1-F(x) =: {\rm e}^{-H(x)}$中的一个参数，其中$H(x)=-\ln(1-F(x))$表示一个正则变化累积风险函数，其正则变化指数等于1/$θ$，$θ\in\mathbb{R}^{+}$。文献中常用的WTC估计量通常定义为对数超出量的函数，因此它们与Pareto型尾部的极值指数（EVI）估计量密切相关。对于正的EVI，经典估计量是Hill估计量。广义均值已成功用于估计EVI，在适当的阈值下可减少偏差和均方根误差。在本研究中，我们在二阶框架内基于对数超出量的幂$p$提出并研究新的WTC估计量类别。通过大规模蒙特卡洛模拟研究评估这些新估计量的性能，并与文献中现有的WTC估计量进行比较。
英文摘要：
The Weibull tail-coefficient (WTC) plays a crucial role in extreme value statistics when dealing with Weibull-type tails. Several distributions, such as normal, Gamma, Weibull, and Logistic distributions, exhibit this type of tail behaviour. The WTC, denoted by $θ$, is a parameter in a right-tail function of the form $ \bar F(x) :=1-F(x) =: {\rm e}^{-H(x)}$, where $H(x)=-\ln(1-F(x))$ represents a regularly varying cumulative hazard function with an index of regular variation equal to 1/$θ$, $θ\in\mathbb{R}^{+}$. The commonly used WTC-estimators in literature are often defined as functions of the log-excesses, making them closely related to estimators of the extreme value index (EVI) for Pareto-type tails. For a positive EVI, the classical estimator is the Hill estimator. Generalized means have been successfully employed in estimating the EVI, leading to reduction of bias and of root mean square error for appropriate threshold values. In this study, we propose and investigate new classes of WTC-estimators based on power $p$ of the log-excesses within a second-order framework. The performance of these new estimators is evaluated through a large-scale Monte-Carlo simulation study, comparing them with existing WTC-estimators available in the literature.
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