Nonparametric inference and Bayesian computing

到目前为止，我开发的非参数（在我之前的发现授权期间）在约束函数的推理上下文中表现得很好。这里感兴趣的函数本质上是受限的，因为它们表征了一个联结，对于一个分布函数，有均匀的边界。copula非常重要，因为它包含随机向量变量之间的整个依赖结构，因此在许多假设检验中很自然地出现。当它们与边际相结合时，它们成为建模联合分布的有力工具。依赖性结构的许多推理方法已经在许多应用科学中找到了具体的应用，例如遗传学和计量经济学。推理方法通常是频率的，在许多情况下是非参数的。在后一种设置中，与联结过程相关的过程（例如肯德尔过程或经验联结过程）是主要的工作工具。这些通常不会导致有限样本的内禀估计，并且经常需要修改它们以满足约束，这是一个不便之处。另一种方法是通过进一步利用问题的几何结构来构建模型，并提供无限维的模型，但仍然易于处理。在这个方向上继续，我有两个主要目标，为未来几年。

The nonparametrics that I developed (during the period of my previous discovery grant), so far in the context of inference on constrained functions, has shown to work well. The functions of interest here are constrained essentially because they characterize a copula, being for one a distribution function, with uniform margins. Copulas are of fundamental importance because they hold the entire dependency structure between the variates of a random vector, and so they appear naturally in many hypothesis tests. When they are combined with marginals, they become a powerful tool for modelling joint distributions. Much of the inferential methodology for the dependence structure has found concrete applications in many applied sciences, such as genetics and econometrics. The inferential approach is usually frequentist, and in many cases nonparametric. In the latter setup, processes linked to the copula (e.g the Kendall process, or empirical copula process) are the main working tools. These do not usually lead to intrinsic estimators for finite samples, and one often needs to modify them in order to satisfy the constraints, an inconvenience here. Another approach is to construct models by further exploiting the geometry of the problem and providing infinite dimensional models, still remaining tractable. Continuing in this direction, I have two main objectives for the next years.