极值理论

M. L. Bianchi, Stoyan Stoyanov, G. Tassinari, Frank J. Fabozzi, S. Focardi
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引用次数: 0

摘要

本章涵盖的主要主题是:极值理论是什么以及它与经典统计学的区别;极值理论的两大支柱:Fisher-Tippett-Gnedenko定理和Pickands-Balkema-de Haan定理;最大值的极限分布将分为三类:Frechet、Weibull或Gumbel分布;广义Pareto分布;极值分布的最大吸引域和尾等价的概念;平稳过程的极大值理论;多元分布的极值理论;多元极值理论中联结的作用;最大似然估计法、矩量法和特殊估计器;估计分布形状参数的Hill估计器和Pickands估计器;通过检查假设分布的线性图的偏差程度来验证统计假设的分位数图(QQ-plot)的使用和局限性;计算众所周知的风险度量(VaR和AVaR)的三种不同方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Extreme Value Theory
The main topics covered in this chapter are:what extreme value theory is and how it differs from classical statistics;the two pillars of extreme value theory: Fisher–Tippett–Gnedenko theorem and Pickands–Balkema–de Haan theorem;the three classes that the limit distribution of maxima will fall into: the Frechet, Weibull, or Gumbel distribution;the generalized Pareto distribution;the maximum domain of attraction of an extreme value distribution and the concept of tail equivalence;the theory of maxima for stationary processes;extreme value theory for multivariate distributions;the role of copula in multivariate extreme value theory;the three types of copulas;three estimation methods for distributions: maximum likelihood estimation method, method of moments, and special estimators;the Hill estimator and the Pickands estimator for estimating the shape parameter of a distribution;use and limitations of the quantile plot (QQ-plot) for verifying statistical hypotheses by examining the degree of deviations of the linearity plot of a hypothesized distribution;three different approaches to compute widely-known risk measures (VaR and AVaR).
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