An elementary approach to extreme values theory
Philippe Barbe
本文提出了一种相当直观的方法来研究极值理论。设计这种方法主要是出于教学的原因。
This note presents a rather intuitive approach to extreme value theory. This approach was devised mostly for pedagogical reason.
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Conditional extreme value models have been introduced by Heffernan and Resnick (2007) to describe the asymptotic behavior of a random vector as one specific component becomes extreme. Obviously, this class of models is related to classical multivariate extreme value theory which describes the behavior of a random vector as its norm (and therefore at least one of its components) becomes extreme. However, it turns out that this relationship is rather subtle and sometimes contrary to intuition. We clarify the differences between the two approaches with the help of several illuminative (counter)ex…
Products in Conditional Extreme Value Model
经典的多变量极值理论试图在多变量吸引区域条件下捕捉各分量之间的极值依赖关系，它要求每个分量也在单变量极值分布吸引区域中。多元极值（MEV）模型有着丰富的理论基础，但在渐近独立的情况下，MEV模型无法捕捉到相关结构，存在一定的局限性。Hynnan和Tawn（2004年）给出了一种不同的MEV方法，他们通过将其中一个分量作为极端条件来检查MEV分布。在这里，我们假设一个组件是在Frechét或威布尔吸引区域力和研究的产品的组件下的条件极值模型的行为。
Data driven extreme value distribution estimation: Derivation of the Mean Integrated Squared Error, optimal bandwidth selection and stability conditions
本文提出了一种数据驱动的极值分布估计器，推导了其均方误差，用于计算最优带宽并建立了带宽优化过程的稳定性条件。
We exploit the asymptotic normality of the extreme value theory (EVT) based estimators of the parameters of a symmetric Lévy-stable distribution, to construct confidence intervals. The accuracy of these intervals is evaluated through a simulation study.
Extreme-Value Copulas
极值Copula函数作为独立随机样本中各分量极大值的Copula函数的极限，可以为稀有事件间的相关结构提供合适的模型。极值Copula不仅在极值理论领域中自然出现，而且可以方便地用于建模一般的正相关结构。本调查的目的是向读者介绍通过极值copula进行依赖建模的最新技术。概率和统计问题进行审查，在非参数以及参数的背景下。
We derive a closed-form solution for the Kullback-Leibler divergence between two Fréchet extreme-value distributions. The resulting expression is rather simple and involves the Euler-Mascheroni constant.
Metastatistics of Extreme Values and its Application in Hydrology
我们提出了一种新的统计处理，“极端事件的元统计”，计算极端事件的频率。这种方法具有普遍有效性，是解决统计不均匀数据问题的适当统计框架。通过使用人工序列，我们表明，元统计产生正确的预测，而传统的方法的基础上广义极值分布不。应用元统计方法的极端事件的情况下，降雨日降水量。
Analysis of the rare and extreme values through statistical modeling is an important issue in economical crises, climate forecasting, and risk management of financial portfolios. Extreme value theory provides the probability models needed for statistical modeling of the extreme values. There are generally two ways to identifying the extreme values in a data set, the block-maxima and the peak-over threshold method. The block-maxima method uses the Generalized Extreme Value distribution and the peak-over threshold method uses the Generalized Pareto distribution. It is common that the location of…
Adjusted inference for multiple testing procedure in group sequential designs
针对成组序贯设计中同时调整多重假设检验与多次分析的研究空白，提出调整序贯p值及交集假设序贯p值，通过加权Bonferroni和加权参数检验控制族系I类错误率进行推断。
Extreme Value Statistics for Analysing Simulated Environmental Extremes
本文介绍Uniofbathtopia团队在EVA2023数据挑战中结合极值理论与无监督学习的方法，并展示了其在模拟环境极值数据上的应用与获胜成果。
The generalized extreme value distribution and its particular case, the Gumbel extreme value distribution, are widely applied for extreme value analysis. The Gumbel distribution has certain drawbacks because it is a non-heavy-tailed distribution and is characterized by constant skewness and kurtosis. The generalized extreme value distribution is frequently used in this context because it encompasses the three possible limiting distributions for a normalized maximum of infinite samples of independent and identically distributed observations. However, the generalized extreme value distribution m…
In extreme value analysis, the extreme value index plays a vital role as it determines the tail heaviness of the underlying distribution and is the primary parameter required for the estimation of other extreme events. In this paper, we review the estimation of the extreme value index when observations are subject to right random censoring and the presence of covariate information. In addition, we propose some estimators of the extreme value index, including a maximum likelihood estimator from a perturbed Pareto distribution. The existing estimators and the proposed ones are compared through a…
A dynamic extreme value model with applications to volcanic eruption forecasting
本文将用于金融风险预测的动态极值模型适配到火山喷发预测，并通过皮通德拉富尔内兹火山案例验证了基于EVT阈值的建模价值。
Similar to the generalized extreme value (GEV) family, the generalized extreme value distributions under power normalization are introduced by Roudsari (1999) and Barakat et al. (2013). In this article, we study the asymptotic behavior of GEV laws under power normalization and derive expressions for the kth moments, entropy, ordering in dispersion, rare event estimation and application of real data set. We also show that, under some conditions, the Shannon entropy and variance of GEV families are ordered.
General extreme value modeling and application of bootstrap on rainfall data - A case study
极值理论研究的是与随机变量序列和随机过程中非常高或非常低的值有关的概率和统计问题。这门学科有着丰富的数学理论，也有着在各个领域应用的悠久传统。在许多关于这个问题的优秀书籍中，科尔斯[2]而这本书集中在数据分析和极端情况的统计推断上。在这篇文章中，我们提出了一个案例研究，其中我们模拟年最大年降雨量数据使用广义极值分布。同时，我们使用R软件进行数据分析，并在附录中给出了R代码.
Elementary proofs of several results on false discovery rate
本文收集了文献中关于BH方法、Benjamini-Yekutieli校正及e-BH方法错误发现率的四个自洽初等证明，并作为推论导出了Simes和Hommel不等式。
This is an elementary introduction to infinite-dimensional probability. In the lectures, we compute the exact mean values of some functionals on C[0,1] and L[0,1] by considering these functionals as infinite-dimensional random variables. The results show that there exist the complete concentration of measure phenomenon for these mean values since the variances are all zeroes.
We show that the set of $d$-variate symmetric stable tail dependence functions, uniquely associated with exchangeable $d$-dimensional extreme-value copulas, is a simplex and determine its extremal boundary. The subset of elements which arises as $d$-margins of the set of $(d+k)$-variate symmetric stable tail dependence functions is shown to be proper for arbitrary $k \geq 1$. Finally, we derive an intuitive and useful necessary condition for a bivariate extreme-value copula to arise as bi-margin of an exchangeable extreme-value copula of arbitrarily large dimension, and thus to be conditionall…
We derive a new (lower) inequality between Kendall's tau? and Spearman's rho? for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.
Improving extreme value statistics
极值统计量的一致收敛性速度是非普适的，可以任意慢。此外，相对误差在近似的尾部可能是无界的，导致难以外推超出可用数据的极值拟合。我们表明，通过使用简单的非线性变换的极值近似可以呈现快速收敛的批量，和渐近的尾巴，从而解决这两个问题。变换通常仅由一个参数参数化，该参数可以数值估计。经典的极值方法被证明是所提出的方法的一个特殊情况。我们证明，可以得到大大改善的结果，几乎没有额外的成本。
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