Estimation and Inference Under Shape Restrictions

在经济学和相关学科的实证研究中，分析关系的标准方法对先验关系的形式施加了强烈的假设。虽然这些假设大大简化了所需的统计方法，但如果它们不正确，得出的结论可能会产生误导。另一种选择是使用所谓的非参数方法，它施加较弱的假设，但结果往往不够精确，无法得出强有力的结论。这项研究探索了使用形状限制来施加额外的结构，但没有指定特定的关系。形状限制，如单调性或凸性，通常是合理的假设，它们可以由基本的经济理论隐含。例如，一种产品的需求在价格上下降。该项目提出了统计方法来估计关系和总结形状限制下的不确定性，这保持了非参数方法的灵活性，但可以得出更精确的结论。这些方法在广泛的现实世界应用中具有吸引力。在本研究中，它们被应用于分析需求函数，在拍卖模型中进行推理，以及估计分位数函数。在技术层面上，该项目克服了由于使用形状限制而导致的统计理论方面的挑战。具体来说，受限估计量的分布取决于形状约束的绑定位置，这是先验未知的。本文提出了一种基于测试反演的推理方法，该方法对未知参数向量或函数产生一致有效的置信区域。该方法适用于广泛的有限维和非参数问题，各种非参数估计，以及许多不同的形状限制。推理基于形状受限估计量的分布，形状受限估计量是无限制估计量在受限参数空间上的近似二次投影，该投影依赖于权矩阵。研究者研究了权矩阵的最优选择，这对受限估计量的性质和相应的置信区域有很大的影响。除了证明均匀的尺寸控制外，该项目还研究了功率特性，以量化使用形状限制所获得的收益。最后，本研究通过研究单调性下的需求模型、参数依赖支持的拍卖模型和非交叉条件分位数函数，说明了这些成果及其在蒙特卡罗模拟和各种经验应用中的广泛适用性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Standard approaches to analyzing relationships in empirical research in economics and related disciplines impose strong assumptions on the form of the relationships a priori. While these assumptions greatly simplify the statistical methods needed, if they are incorrect, conclusions obtained can be misleading. An alternative is to use so called nonparametric methods, which impose weaker assumptions, but consequently, the results obtained are often not precise enough to reach strong conclusions. This research explores the use of shape restrictions to impose additional structure, but without specifying a particular relationship. Shape restrictions, such as monotonicity or convexity, are often reasonable assumptions and they can be implied by basic economic theory. For example, the demand of a product is decreasing in its price. The project proposes statistical methods to estimate relationships and summarize uncertainty under shape restriction, which keep the flexibility of nonparametric approaches, but can yield much more precise conclusions. These methods are appealing in a wide range of real world applications. In this research, they are applied to analyze demand functions, conduct inference in auctions models, and estimate quantile functions. At a technical level the project overcomes challenges in the statistical theory resulting from the use of shape restrictions. Specifically, the distribution of the restricted estimator depends on where the shape restrictions bind, which is unknown a priori. This research suggests an inference method based on test inversion, which yields uniformly valid confidence regions for unknown parameter vectors or functions. The method applies to a wide range of finite dimensional and nonparametric problems, to various nonparametric estimators, and to many different shape restrictions. Inference is based on the distribution of a shape restricted estimator, which is an approximate quadratic projection of an unrestricted estimator onto a restricted parameter space and this projection depends on a weight matrix. The investigator studies optimal choices of the weight matrix, which can have large effects on properties of the restricted estimators and the corresponding confidence regions. Next to proving uniform size control, the project also investigates power properties to quantity the gains from using shape restrictions. Finally, this research illustrate these gains and the wide applicability in Monte Carlo simulations and in various empirical applications, studying demand models under monotonicity, auction models with parameter dependent support, and non-crossing conditional quantile functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.