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

了解高维的极值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。 当相关性由协方差描述时，存在许多统计方法来表征和建模高维数据的结构。 然而，协方差是一个差的描述分布的联合尾部和方法，专门设计的极端是必要的联合风险的准确量化。 虽然用于描述极值依赖的理论上合理的框架是已知的，但实践者非常需要用于高维极值的统计方法。 基于研究者以前的工作，这个项目将提出并发展一个新的极端多元分布的属性。 该分布的特征在于一个参数矩阵，它总结了成对的尾部依赖性，如协方差矩阵，但它与一个理论上合理的极端框架相关联。 这种分布，再加上由研究者开发的工具，将允许从业者建模和表征金融，保险或气象应用中产生的高维数据的风险。该项目还将涉及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，最近关于极值的变换线性模型的工作，以及通过尾部成对依赖矩阵（TPDM）表征极值依赖的工作，已经在极值建模和传统线性统计方法之间建立了联系。极值类似主成分分析，空间自回归模型，线性自回归移动平均（阿尔马）时间序列模型，线性预测，偏相关已经建成。然而，参数估计到目前为止已经有点ad-hoc，并已基于最小化模型的TPDM值和经验估计之间的平方差。 本计画提出一种新的机率分布，即以TPDM为参数的转换线性T分布。 由于该分布具有封闭形式的密度，因此它使得TPDM的似然估计成为可能。 此外，该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，因为经典阿尔马模型的因果类似物显示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相关工作，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.