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

非参数估计和推理：形状约束、模型选择和水平集估计

基本信息

批准号：
1712664
负责人：
Charles Doss
金额：
$ 9.98万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712664&HistoricalAwards=false
关键词：
Nonparametric Estimation Inference Shape Constraints

项目摘要

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.

更大和更复杂的数据集变得越来越普遍。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事先假设。因此，灵活的方法是必要的，在现代景观，但他们有困难，从业者一般必须“调整”的方法，以获得可靠的结果。这给数据分析引入了一个特殊的元素，导致结果缺乏可复制性，并且意味着（不正确调整的）统计程序可能会返回错误的结果。该项目的统一主题是开发非常灵活且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。该项目的应用领域多种多样，包括疫苗试验分析、经济数据研究和异常值检测问题（广泛应用于金融服务）。非常灵活的非参数统计方法已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（可能有许多）调谐参数，这些参数必须选择得很好以确保可靠的性能。本提案的重点是开发可实施的方法，并产生可靠的结果，而不需要最终用户采取任何特别步骤。将研究三个主要问题：（a）形状约束估计的模型选择，（B）形状约束估计的似然比类型检验，（c）多元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，其具有同时是非参数的但也具有自动选择最佳调谐参数的益处。此外，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）在许多情况下，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（例如水平集流形，由离群值检测问题激发）。当形状约束不适用时，将考虑形状约束方法和替代方法。由于这些问题的非标准性质，在所考虑的许多问题中几乎没有或没有有效的程序。