Nonparametric inference and Bayesian computing

到目前为止，我在关于受约束函数的推理的上下文中开发的非参数方法(在我之前的发现授权期间)已经被证明工作得很好。这里感兴趣的函数本质上是受限制的，因为它们的特征是Copula，其中一个是分布函数，具有一致的边距。联结具有基本的重要性，因为它们包含随机向量的变量之间的整个依赖结构，因此它们在许多假设检验中自然出现。当它们与边际相结合时，它们就成为模拟节理分布的强大工具。相依性结构的许多推理方法在许多应用科学中都有具体的应用，例如遗传学和计量经济学。推理方法通常是频繁的，在许多情况下是非参数的。在后一种设置中，与Copula相关联的过程(例如，Kendall过程或经验Copula过程)是主要的工作工具。这些通常不会导致有限样本的本征估计，人们经常需要修改它们以满足约束，这在这里是不方便的。另一种方法是通过进一步利用问题的几何形状并提供仍然易于处理的无限维模型来构建模型。继续朝着这个方向前进，我对未来几年有两个主要目标。 我的第一个目标是研究我在2维研究的筛子模型的扩展。我想探索贝叶斯的可能性，其优点基本上是预测，其中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，我总是假设边际分布是已知的，或者它们属于某个筛子家族。我的第二个目标是取消所有关于仅使用等级的边际分布的假设。由于提案中讨论的两个主要原因，在推断依赖结构时，与职级合作是很自然的。本质上，这里(边距)的讨厌参数的影响被消除了，特别是边距上的先验对后验没有影响。一个主要的(具有挑战性的)目标是计算排序可能性。虽然这里目前正在开发渐近性，但我将专注于通过快速混合马尔科夫链来构造有限样本的良好随机逼近。 只使用等级的频繁竞争者是经验系词(它不是真正的系词)。此外，极限高斯场过程的协方差结构复杂，必须采用自举近似。我提出了一个基于模型的估计量，它具有相同的限制过程，但希望证明它具有比经验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.