Copula modeling with generative neural networks

我建议的研究计划福尔斯落在Copula建模和计算统计与定量风险管理（QRM）的应用领域。Copula建模是关于具有连续边缘分布的随机向量的分量之间的依赖性的建模。应用程序通常需要从底层Copula模型中进行采样，例如，用于计算QRM中的风险度量，金融和保险中的定价应用，或者在统计应用中计算罕见事件概率。维数越大，就越难为给定的数据找到合适的copula模型。从实践的角度来看，灵活的和潜在的高维copula模型是需要的，可以很容易地拟合到数据，并快速采样。为了在估计诸如上述的量时从方差减小中受益，人们还希望具有来自相应模型的准随机数生成器（QRNG），即，从随机化的准蒙特卡罗点集到相应copula样本的差异保持变换。然而，有许多copula模型，没有或没有有效的QRNG。为此，我的研究项目主要集中在生成神经网络（GNNs），特别是生成矩匹配网络（GMMN）。我们在这个新方向的开创性工作利用GMMN从任意copula模型构建QRNG。特别吸引人的是这种方法的普遍性和可计算性。我提出的研究计划的首要目标是确定GNNs（如GMMN）在多大程度上可以解决经典参数模型的局限性。目标沿着这种方式如下：在我们最初的工作中，我们确定了三种情况下，可能会导致问题的能力GMMN正确学习的基础分布和目标是解决这些情况。我们还发现，随着维数的增加，低差异属性似乎会恶化，我们计划进一步研究。对于企业来说，一个重要的目标是拥有有意义的统计数据和图形工具来总结和比较GNN。另一个目标是研究是否可以利用GMMN来构建拟合优度检验。我们还旨在开发用于在R中建模涉及GNN的任务的算法和函数，这允许任何人复制和应用我们的研究。我们的最终目标是将我们的研究结果应用于QRM中具有挑战性的问题，例如（系统性）风险度量和资本配置的估计，其中GMMN为各种各样的模型提供了一种有前途的新方法。研究生将是这个研究计划的一个组成部分。他们将获得统计和概率的知识，对高维依赖模型的构建和挑战的理解，以及计算技能，包括应用神经网络解决实际相关依赖问题的能力。

My proposed research program falls in the area of copula modeling and computational statistics with applications to quantitative risk management (QRM). Copula modeling is concerned with the modeling of the dependence between the components of a random vector with continuous marginal distributions. Applications typically require sampling from the underlying copula model, for example, for computing risk measures in QRM, pricing applications in finance and insurance, or when computing rare-event probabilities in statistical applications. The larger the dimension, the more difficult it is to find an adequate copula model for given data. From a practical point of view, flexible and potentially high-dimensional copula models are needed that can easily be fitted to data and that are fast to sample from. To benefit from variance reduction when estimating quantities such as those above, one would also like to have a quasi-random number generator (QRNG) from the respective model, that is, a discrepancy-preserving transformation from a randomized quasi-Monte Carlo point set to the respective copula sample. However, there are many copula models for which there is no or no efficient QRNG. To this end, my research program focuses on generative neural networks (GNNs), in particular, generative moment matching networks (GMMNs). Our pioneering work in this new direction utilized GMMNs to construct QRNGs from an arbitrary copula model. Particularly appealing are the universality and the computability of this approach. The overarching goal of my proposed research program is to determine to what extent GNNs such as GMMNs can address the limitations of classical parametric models. Objectives along this way are the following: In our initial work we identified three scenarios which can cause problems in terms of the ability of GMMNs to properly learn the underlying distributions and a goal is to address these scenarios. We also found that the low discrepancy property seems to deteriorate for increasing dimension which we plan to investigate further. A goal important for businesses is to have meaningful statistics and graphical tools to summarize and compare GNNs. Another goal is to investigate whether GMMNs can be utilized to construct goodness-of-fit tests. We also aim at developing algorithms and functions for modeling tasks involving GNNs in R which allows anyone to reproduce and apply our research. Our final goal is to apply our findings to challenging problems in QRM such as the estimation of (systemic) risk measures and capital allocations, where GMMNs provide a promising new approach for a wide variety of models. Graduate students will be an integral part of this research program. They will gain knowledge of statistics and probability, an understanding of the construction and challenges of high-dimensional dependence models, as well as computational skills including the ability to apply neural networks to solve practically relevant dependence problems.