Foundations of High-Dimensional and Nonparametric Hypothesis Testing

统计推断工具是统计学科向经验科学的主要输出，是自然科学家解释观测结果和量化其结论的不确定性的主要透镜。然而，在现代大型数据集的分析中，我们可用的最常见的推理工具充满了陷阱，通常需要在有效应用之前检查各种技术条件。这反过来又导致了对推理工具的滥用和对结果的误解。该研究项目旨在通过开发和分析新的用户友好的方法来解决这一问题，以便在复杂的环境中进行统计推断。我们开发的方法将广泛适用于物理和生物科学中各种具有挑战性的推理问题，将消除验证技术条件的需要，并最终在其应用中具有强大的功能。主要和共同主要的研究人员将参与指导和指导研究生，在课程和课程的开发，并在整合项目与研究小组的统计方法在物理科学（STAMPS）。这个项目将推进我们的理解，高维和非参数推理沿着三个前沿。首先，我们的目标是开发统计推断工具的不规则模型，这是有效的弱条件下。我们将特别关注混合模型，以及使用样本分裂以避免强正则性条件的方法。其次，我们将证明我们的方法以惊人的小统计代价实现了这些强有力的保证。为了严格量化为避免强正则性条件而付出的统计代价，我们将使用极大极小理论。然而，在许多情况下，标准的极大极小理论并不能充分捕捉统计推断的难度，因为推断的难度在参数空间中可能会有很大的变化。一个更精确的理论--称为局部极小极大理论--可以得到更精确的图像，我们将通过这个透镜来研究我们的方法。最后，我们将讨论条件独立性（CI）测试的问题。尽管它在回归诊断和概率图形模型的研究中发挥着核心作用，但CI测试的任务及其内在困难却知之甚少。我们将解决CI测试的两个基本方面，通过研究方法，以适当地校准CI测试，并通过开发和分析强大的新CI测试。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Statistical inferential tools are the main export from the discipline of statistics to the empirical sciences, serving as the primary lens through which natural scientists interpret observations and quantify the uncertainty of their conclusions. However, in the analysis of modern large datasets the most common inferential tools available to us are fraught with pitfalls, often requiring various technical conditions to be checked before their valid application. This in turn has led to misuse of the inferential tools and subsequent misinterpretation of results. This research project will aim to address this issue by developing and analyzing new user-friendly methodologies for statistical inference in complex settings. The methods we develop will be broadly applicable to a wide variety of challenging inferential problems in the physical and biological sciences, will eliminate the need to verify technical conditions, and will ultimately be robust in their application. The principal and co-principal investigators will be involved in advising and mentoring graduate students, in curricular and course development, and in integrating the project with a research group on Statistical Methods in the Physical Sciences (STAMPS).This project will advance our understanding of high-dimensional and non-parametric inference along three frontiers. Firstly, we aim to develop statistical inferential tools for irregular models, which are valid under weak conditions. Our particular focus will be on mixture models, and on methods which use sample-splitting to avoid strong regularity conditions. Secondly, we will show that our methods achieve these strong guarantees at a surprisingly small statistical price. To rigorously quantify the statistical price paid for avoiding strong regularity conditions we will use minimax theory. However, standard minimax theory, in many cases, does not adequately capture the difficulty of statistical inference since the difficulty of inference can vary significantly across the parameter space. A more refined theory -- called local minimax theory -- leads to a more accurate picture, and we will study our methods via this lens. Finally, we will address the problem of conditional independence (CI) testing. Despite its central role in regression diagnostics, and in the study of probabilistic graphical models, the task of CI testing and its intrinsic difficulty is poorly understood. We will address two fundamental aspects of CI testing, by studying methods to appropriately calibrate CI tests, and by developing and analyzing powerful new CI tests.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.