Multivariate Dependence Modeling with Copulas

建议的研究集中在基于Copula的依赖模型的新的统计方法上，该模型现在被广泛应用于许多领域，包括环境和健康科学、金融、保险和风险管理。多变量数据中的相关性是一种复杂的现象，基于Copula的模型比传统的多变量分布更能解释这种现象。我的研究计划是我之前NSERC发现资助的延续，新的重点是复杂的数据结构和最近应用程序中的挑战，例如高维数据、不连续变量或低概率、高影响事件。 迫切需要通用性强且易于处理的多变量Copula模型，这些模型非常适合于高维数据的处理。我将设计这种使用随机表示法的模型，为它们开发统计方法，并通过统计计算R项目向统计的最终用户提供这种新的方法。我还将使用新模型来构建集群数据的分层依赖结构。我研究的另一部分将集中于极端事件之间的相关性建模：我将开发基于约束B-样条估计的新统计技术；估计器将对应于真正的模型，从这些模型可以模拟和预测灾难性事件。此外，我还将进一步研究非连续数据的经验Copula过程。这项工作将作为基于Copula的离散、混合或不连续变量模型的有效统计方法的基础。这些新技术还将在变量数量较多的情况下进行研究；其目的将是设计图形工具，以可视化高维数据中的相关性。我的方法论工作将应用于各种实际问题，例如，水文学、保险和风险管理。 这项工作的影响将是改变研究人员对多变量数据建模的方式。它将展示基于Copula的技术的实用性，以及它即使对于复杂的数据结构也是可行的。这项研究将有助于不断增长的高维依赖建模的文献，这是一个广泛的统计受众感兴趣的。鉴于与依赖有关的问题在金融、保险、水文学和风险管理等领域的重要性，这项工作将在这些领域产生平行影响。将通过统计计算R项目向从业人员提供新的方法。

The proposed research focuses on new statistical methodology for copula-based dependence models, which are now widely used in many fields, including environmental and health sciences, finance, insurance, and risk management. Dependence in multivariate data is a complex phenomenon and copula-based models are far more capable of accounting for it than traditional multivariate distributions. My research program is a continuation of my previous NSERC Discovery Grant, with a new focus on complex data structures and recent challenges in applications, such as high-dimensional data, discontinuous variables, or low-probability, high-impact events. There is a pressing need for versatile yet tractable multivariate copula models that are well suited for the treatment of high-dimensional data. I will devise such models using stochastic representations, develop statistical methodology for them, and make this new methodology available to end-users of statistics through the R Project for Statistical Computing. I will also use the new models to build hierarchical dependence structures for clustered data. Another part of my research will focus on the modeling of dependence between extreme events: I will develop novel statistical techniques based on constrained B-spline estimation; the estimators will correspond to genuine models from which it is possible to simulate and predict catastrophic events. Moreover, I will further my work on empirical copula processes for non-continuous data. This work will serve as a basis for valid statistical methodology for copula-based models for variables that are discrete, mixed or otherwise discontinuous. These new techniques will also be investigated in settings where the number of variables is large; the aim will be to devise graphical tools for the visualization of dependence in high-dimensional data. My methodological work will be applied to various practical problems, e.g., in hydrology, insurance, and risk management. The impact of the work will be to change the manner in which researchers model multivariate data. It will demonstrate the usefulness of copula-based techniques, and its feasibility even for complex data structures. The research will contribute to the growing literature on high-dimensional dependence modeling, which is of interest to a wide statistical audience. Given the importance of dependence-related issues in fields such as finance, insurance, hydrology, and risk management, the work will have parallel impact in these areas. New methodology will be made accessible to practitioners through the R Project for Statistical Computing.