A new parametric model, likelihood methods, and other advancements for multivariate extremes

了解高维度的极端依赖对于量化包括环境和气候科学在内的各种学科中多种因素的组合所产生的风险至关重要。当相关性用协方差来描述时，有许多统计方法可以用来刻画高维数据的特征和建模结构。然而，协方差不能很好地描述分布的联合尾部，而专门为极端情况设计的方法对于准确量化联合风险是必要的。虽然在理论上证明了描述极端相关性的框架是已知的，但高维极端情况的统计方法是实践者非常需要的。在研究人员以前工作的基础上，这个项目将展示和开发一种新的极端情况下的多变量分布的性质。这种分布的特征是一个参数矩阵，它总结了像协方差矩阵一样的成对尾部依赖关系，但它链接到一个理论上合理的极值框架。这种分布，再加上研究人员正在开发的工具，将允许从业者对金融、保险或气象应用中出现的高维数据的风险进行建模和表征。该项目还将培训一名研究生进行极值分析，并与政府实验室的大气科学家合作。更详细地说，最近在极值转换线性模型以及通过尾对相关矩阵(TPDM)表征极值相关性的工作中，在极值建模和传统线性统计方法之间建立了联系。构造了主成分分析、空间自回归模型、线性自回归滑动平均(ARMA)时间序列模型、线性预测和偏相关的极值模拟。然而，到目前为止，参数估计在某种程度上是特别的，并且一直基于最小化模型的TPDM值和经验估计之间的平方差。本课题提出了一种新的概率分布--变换线性T分布，它以TPDM值为参数。由于该分布具有闭合形式的密度，因此可以对TPDM值进行似然估计。此外，这个项目将扩展研究人员最近的线性时间序列工作，以建立非因果模型，因为与经典ARMA模型的因果相似显示出数据中没有看到的不对称性。该项目还将扩展最近的部分尾部关联工作，为图形模型添加因果方向。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Understanding extremal dependence in high dimensions is essential for quantifying risk arising from a combination of multiple factors in a variety of disciplines including the environmental and climate sciences. When dependence is described by covariance, many statistical methods exist to characterize and model structure for high-dimensional data. Covariance, however, is a poor descriptor of a distribution's joint tail and methods specifically designed for extremes are necessary for accurate quantification of joint risk. While theoretically-justified frameworks for describing extremal dependence are known, statistical methods for high dimensional extremes are very much needed by practitioners. Building on the investigator's previous work, this project will present and develop the properties of a new multivariate distribution for extremes. The distribution is characterized by a parameter matrix which summarizes pairwise tail dependencies like a covariance matrix, but which is linked to a theoretically-justified framework for extremes. This distribution, coupled with tools in development by the investigator, will allow a practitioner to model and characterize risk for high dimensional data arising in finance, insurance, or meteorological applications. The project will also involve training a graduate student in extreme value analysis and collaboration with atmospheric scientists in government labs.In more detail, recent work on transformed-linear models for extremes coupled with characterizing extremal dependence via the tail pairwise dependence matrix (TPDM) has built connections between extremes modeling and traditional linear statistics methods. Extremal analogues to principal component analysis, spatial autoregressive models, linear autoregressive moving average (ARMA) time series models, linear prediction, and partial correlation have been constructed. However, parameter estimation has thus far been somewhat ad-hoc, and has been based minimizing squared differences between the model's TPDM values and empirical estimates. This project presents a new probability distribution, the transformed-linear T-distribution, which has the TPDM as a parameter. As this distribution has a closed-form density, it makes likelihood estimation of the TPDM possible. Additionally, this project will extend the investigator's recent linear time series work to build non-causal models, as the causal analogs to classical ARMA models show an asymmetry not seen in the data. This project will also extend the recent partial tail correlation work to add causal direction to the graphical models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.