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强加了强有力的假设。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，所得出的结论可能会产生误导。另一种方法是使用所谓的非参数方法，该方法施加较弱的假设，但因此，获得的结果往往不够精确，无法得出强有力的结论。本研究探讨了使用形状限制强加额外的结构，但没有指定一个特定的关系。形状限制，如单调性或凸性，往往是合理的假设，它们可以隐含在基本的经济理论。例如，一种产品的需求量在下降。 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，保持了非参数方法的灵活性，但可以产生更精确的结论。这些方法在广泛的真实的世界应用中具有吸引力。在本研究中，他们被应用于分析需求函数，进行拍卖模型的推断，并估计分位数函数。在技术层面上，该项目克服了使用形状限制所带来的统计理论挑战。具体而言，受限估计器的分布取决于形状限制约束的位置，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，该方法可以对未知参数向量或函数产生一致有效的置信域。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。推断是基于形状限制估计的分布，这是一个近似的二次投影的无限制的估计到一个限制的参数空间，这个投影取决于一个权重矩阵。研究者研究了权矩阵的最优选择，这对约束估计量的性质和相应的置信区域有很大的影响。除了证明统一的尺寸控制，该项目还研究了功率特性，以量化使用形状限制的收益。最后，这项研究说明了这些收益和广泛的适用性，在蒙特卡洛模拟和各种实证应用，研究单调性下的需求模型，拍卖模型与参数相关的支持，和非交叉条件分位数functions.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.