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。***我的第一个目标是对我在维度2中研究过的筛子模型进行扩展。我想探索贝叶斯可能性，它的优势在于预测，未来观测的不确定性通过预测密度自然地与参数的不确定性结合在一起。***在我以前的工作中，我总是假设边缘分布是已知的，或者它们属于某些筛子族。我的第二个目标是只使用排名消除对边际分布的所有假设。根据建议中讨论的两个主要原因，在推断依赖结构时使用等级是很自然的。从本质上讲，这里的干扰参数（边缘）的影响被消除了，特别是，边缘的先验对后验没有影响。一个主要的（具有挑战性的）目标是秩似然的计算。虽然渐近性目前正在发展，但我将重点放在通过快速混合马尔可夫链构建有限样本的良好随机近似上。****一个只使用秩的频率主义者竞争者是经验联结（不是真正的联结）。此外，极限高斯场过程的协方差结构复杂，自举近似成为必要条件。我提出了一个基于模型的估计器，它具有相同的极限过程，但希望表明它具有比经验copula更好的有限样本行为。特别是，该模型应该给出更好的自举近似

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. ***My first objective is to work on extensions of sieves models that I have studied in dimension 2. I want to explore Bayesian possibilities, the advantage being essentially prediction, where the uncertainty of the future observations is naturally combined to that of the parameters via the predictive density.***In my previous work, I have always made the assumption that the marginal distributions are known or that they belong to some family of sieves. My second objective consists in removing all assumptions on the marginal distributions using ranks only. It is natural to work with ranks when inferring on the dependence structure for two main reasons discussed in the proposal. Essentially, the effect of the nuisance parameters here (the margins) is eliminated, and in particular, the prior on the margins has no effect on the posterior. A main (challenging) goal is the calculation of the rank-likelihood. While asymptotics are currently being developed here, I will focus on the construction of good stochastic approximations via rapidly mixing Markov chains, for finite samples.****A frequentist competitor that uses ranks only is the empirical copula (which is not a genuine copula). Moreover, the covariance structure of the limiting Gaussian field process is complicated and bootstrap approximations become essential. I propose a model based estimator, that has the same limiting process, but hope to show that it has better finite samples behaviour than the empirical copula. In particular, the model should give better bootstrap approximations.**