Multivariate Dependence Modeling with Copulas

建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。多变量数据中的依赖性是一个复杂的现象，基于公式的模型比传统的多变量分布更能解释它。我的研究计划是我之前的NSERC发现资助的延续，新的重点是复杂的数据结构和应用中的最新挑战，如高维数据、不连续变量或低概率、高影响事件。***迫切需要一种适合处理高维数据的通用但易于处理的多元联结模型。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R统计计算项目将这种新方法提供给统计的最终用户。我还将使用新模型为聚集数据构建分层依赖结构。我的研究的另一部分将集中在极端事件之间的依赖性建模：我将开发基于约束b样条估计的新的统计技术；估计器将对应于真实的模型，从这些模型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联结过程。这项工作将为离散、混合或不连续变量的基于copula模型的有效统计方法奠定基础。这些新技术也将在变量数量很大的环境中进行研究；其目的是设计图形工具，使高维数据的依赖性可视化。我的方法论工作将应用于各种实际问题，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的技术的有用性，以及它对于复杂数据结构的可行性。该研究将有助于高维依赖性建模的文献的增长，这是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。新的方法将通过统计计算R项目提供给实践者

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.***