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

了解高维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组合所产生的风险至关重要。当用协方差来描述相关性时，存在许多统计方法来表征高维数据的结构并对其进行建模。然而，协方差并不能很好地描述分布的联合尾部，而为极端情况专门设计的方法对于准确量化联合风险是必要的。虽然描述极值依赖性的理论框架是已知的，但实践者非常需要高维极值的统计方法。在研究者先前工作的基础上，本项目将呈现并发展一种新的多元极端分布的性质。分布的特征是一个参数矩阵，它像协方差矩阵一样总结了成对的尾部依赖关系，但它与一个理论上合理的极值框架相关联。这种分布与研究者正在开发的工具相结合，将允许从业者对金融、保险或气象应用中出现的高维数据进行建模和表征风险。该项目还包括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，最近关于极值的转换线性模型以及通过尾部成对相关矩阵（TPDM）表征极值相关性的工作已经在极值建模和传统线性统计方法之间建立了联系。构造了主成分分析的极值类似物、空间自回归模型、线性自回归移动平均（ARMA）时间序列模型、线性预测和偏相关。然而，到目前为止，参数估计有些特别，并且基于最小化模型的TPDM值和经验估计之间的平方差。本文提出了一种新的概率分布，即以TPDM为参数的变换线性t分布。由于该分布具有封闭形式的密度，因此可以对TPDM进行似然估计。此外，该项目将扩展研究者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模型的因果类比显示了数据中未见的不对称性。该项目还将扩展最近的部分尾部相关工作，为图形模型添加因果方向。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.