Nonparametric Estimation and Inference: Shape Constraints, Model Selection, and Level Set Estimation

更大、更复杂的数据集正变得越来越普遍。因此，使用非常灵活的统计方法是有利的，也是必要的，这些方法允许数据“为自己说话”，而不是让研究人员对数据做出强烈的、不合理的预先假设。因此，灵活的方法在现代环境中是必要的，但它们有一个困难，即从业者通常必须“调整”方法以获得可靠的结果。这给数据分析引入了一个特别的元素，导致结果缺乏可复制性，并且意味着（未正确调优的）统计过程可能返回不正确的结果。这个项目的统一主题是开发既非常灵活又完全自动化的统计方法，这意味着它们不依赖于用户选择的调优参数。推动这一项目的应用领域是多种多样的，包括疫苗试验分析、经济数据研究和异常值检测问题（广泛用于金融服务）。非常灵活的非参数统计方法已经成为处理大型数据集复杂性的必要工具。在实践中使用非参数工具的一个困难是它们通常依赖于（可能很多）调优参数，必须选择好这些参数以确保可靠的性能。本建议的重点是开发可实施的方法，并在不需要最终用户采取任何特别步骤的情况下产生可靠的结果。将研究三个主要问题：(a)形状约束估计器的模型选择，(b)形状约束估计器的似然比类型检验，以及(c)多变量密度复杂特征的估计和推断。在(a)和(b)中，重点是使用所谓的形状约束估计器，它具有同时是非参数的优点，而且还具有自动选择最优调谐参数的优点。此外，在许多情况下，它们产生于自然或公理化的先验信息（例如，经济理论或物理定律），在这种情况下，人们当然应该使用这些信息。在(c)中，重点是估计密度的复杂特征（例如由离群值检测问题驱动的水平集流形）。当形状约束不适用时，将考虑形状约束方法和替代方法。在考虑的许多问题中，由于问题的非标准性质，很少或没有有效的程序可用。

Larger and more complex data sets are becoming more and more commonplace. It is thus both advantageous and necessary to use statistical methods that are very flexible, and which allow the data to "speak for itself," rather than having researchers make strong unjustifiable prior assumptions about the data. Flexible methods are thus necessary in the modern landscape, but they have the difficulty that the practitioner generally must "tune" the methods in order to get reliable results. This introduces an ad-hoc element to data analysis, leads to a lack of replicability in results, and means (incorrectly tuned) statistical procedures may return incorrect results. The unifying theme of this project is the development of statistical methods that are both very flexible and also fully automated, meaning they do not depending on user-chosen tuning parameters. The application areas motivating this project are varied, and include the analysis of vaccine trials, the study of economic data, and the problem of outlier detection (used widely in financial services).Very flexible nonparametric statistical methods have become necessary tools to handle the complex nature of large data sets. One difficulty with using nonparametric tools in practice is their general dependence on (potentially many) tuning parameters which must be chosen well to ensure reliable performance. The focus of this proposal is on developing methods which can be implemented and lead to reliable results without requiring any ad-hoc steps by the end user. Three main problems will be studied: (a) model selection for shape-constrained estimators, (b) likelihood ratio type tests for shape-constrained estimators, and (c) estimation and inference for complex features of multivariate densities. In (a) and (b) the focus is on using so-called shape-constrained estimators, which have the benefit of simultaneously being nonparametric but also of automatically selecting optimal tuning parameters. Furthermore, they arise out of natural or axiomatic prior information (e.g., economic theory or laws of physics) in many settings, and in such cases one should certainly use that information. In (c), the focus is on estimation of complex features of densities (such as level set manifolds, motivated by outlier detection problems). Both shape-constrained methods and alternative methods will be considered, when shape constraints are not applicable. There are few or no effective procedures available in many of the problems under consideration, because of the nonstandard nature of the problems.