Multivariate Dependence Modeling with Copulas

拟议的研究重点是基于Copula的依赖模型的新统计方法，这些模型现在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。多元数据的相关性是一个复杂的现象，基于Copula的模型比传统的多元分布更能解释它。我的研究计划是我以前的NSERC发现补助金的延续，新的重点是复杂的数据结构和应用程序中的最新挑战，如高维数据，不连续变量，或低概率，高影响事件。 有一个通用的，但易于处理的多元copula模型，非常适合于处理高维数据的迫切需要。我将使用随机表示设计这样的模型，为它们开发统计方法，并通过统计计算R项目将这种新方法提供给统计的最终用户。我还将使用新的模型为集群数据构建层次依赖结构。我的研究的另一部分将集中在极端事件之间的相关性建模：我将开发基于约束B样条估计的新的统计技术;估计量将对应于真正的模型，从中可以模拟和预测灾难性事件。此外，我将进一步研究非连续数据的经验copula过程。这项工作将作为一个有效的统计方法的基础，基于Copula模型的变量是离散的，混合的或不连续的。这些新技术也将在变量数量很大的情况下进行研究;目的是设计图形工具，使高维数据的相关性可视化。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的技术的有用性，以及它甚至对于复杂数据结构的可行性。这项研究将有助于不断增长的文献高维依赖建模，这是感兴趣的广泛的统计观众。鉴于依赖相关问题在金融、保险、水文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.