Copula modeling with generative neural networks

我建议的研究项目属于Copula建模和计算统计领域，以及量化风险管理(QRM)的应用。Copula建模涉及对具有连续边缘分布的随机向量的分量之间的相关性进行建模。应用程序通常需要从基础Copula模型中进行采样，例如，在QRM中计算风险度量，在金融和保险中为应用程序定价，或者在统计应用程序中计算罕见事件的概率。维度越大，为给定数据找到合适的Copula模型就越困难。从实用的角度来看，需要灵活的和潜在的高维的Copula模型，这种模型可以很容易地与数据拟合，并且采样速度很快。为了在估计如上所述的量时受益于方差减少，人们还希望从相应的模型中获得准随机数生成器(QRNG)，即从随机化的准蒙特卡罗点集到相应的Copula样本的保差变换。然而，有许多Copula模型没有或没有有效的QRNG。 为此，我的研究项目集中在产生式神经网络(GNN)，特别是产生性矩匹配网络(GMMN)。我们在这一新方向上的开创性工作是利用GMMN从任意的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。我提议的研究计划的首要目标是确定GMMN等GNN在多大程度上能够解决经典参数模型的局限性。沿着这条道路的目标如下：在我们的初步工作中，我们确定了在GMMN正确学习基本分布的能力方面可能造成问题的三种情况，目标是解决这些情况。我们还发现，低偏差性质似乎随着维度的增加而恶化，这一点我们计划进一步研究。对企业来说，一个重要的目标是拥有有意义的统计数据和图形化工具来总结和比较GNN。另一个目标是调查GMMN是否可以用于构建拟合优度测试。我们还致力于开发用于对涉及R中GNN的任务进行建模的算法和函数，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发现应用于QRM中具有挑战性的问题，例如(系统性)风险度量的估计和资本分配，其中GMMN为各种模型提供了一种有前途的新方法。 研究生将是这项研究计划不可或缺的一部分。他们将获得统计学和概率的知识，了解高维依赖模型的构建和挑战，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.