Non-asymptotic inference for high and infinite dimensional data

高维和无限维数据的非渐近推理

• 批准号: RGPIN-2018-05678
• 负责人: Kashlak, Adam
• 金额: $ 1.68万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713782
• 关键词: Non; asymptotic; inference; infinite; dimensional

When analyzing high and infinite dimensional data, novel statistical techniques that go beyond classical asymptotic theory are required. One class of such alternatives are the exact and permutation tests that aim to characterize the finite sample distribution of the data [Pigoli et al., 2014, Cabassi et al., 2017]. However, such approaches are often mired in the computational complexity of combinatorial enumeration and thus cannot be fully realized. This research proposal aims to attack the same problem of characterizing the finite sample distribution of the data by the careful construction and implementation of concentration inequalities, the so-called non-asymptotic theory of independence [Boucheron et al., 2013]. There are three facets to this research. The first is investigations into non-asymptotic methods for high dimensional data. The main focus is on covariance and precision matrix estimation for high dimensional data using concentration inequalities and other non-asymptotic tools. The secondary focus is on the applicability of such estimators to actual inferential problems. The second facet is research into log-concave stochastic processes for functional data. The main goal is to carefully construct nonparametric classes of stochastic processes that have enough nice properties--e.g. strong concentration behaviour--to be used for inference on functional data problems including speech analysis and neuroimaging. The final topic is inference on transformation invariant distributions. One problem of interest is to determine a testing methodology for the detection of invariance of data to some transformation. For example, the multivariate Gaussian distribution has ellipsoidal symmetry. A second problem is to investigate concentration and contraction properties of random variables when such transformations are applied. This is an extension of the Rademacher averages, which yield fascinating and highly useful properties. The impact of this research proposal can be quite far reaching. Primarily, it aims to develop methodology, which will be widely applicable to some of the most complex and inscrutable data available to scientific researchers. This includes high dimensional data such as gene expressions and epidemiological studies. It also includes infinite dimensional or functional data such as neuroimaging and other medical imaging. It furthermore can be applied to various types of spatial temporal data such as collections of climate measurements. Beyond such applications, this research will have close connections with topics in probability theory and functional analysis leading to many well cited papers in top journals for both statistical methodology and statistical theory.

在分析高和无限的尺寸数据时，需要超越经典渐近理论的新型统计技术。一类此类替代方案是旨在表征数据的有限样本分布的确切和排列测试[Pigoli等，2014； Cabassi等，2017]。但是，这种方法通常在组合枚举的计算复杂性中陷入困境，因此无法完全实现。这项研究建议旨在通过仔细的构造和浓度不平等的实施，即所谓的非反应独立性理论来攻击数据的有限样本分布[Boucheron等，2013]。 这项研究有三个方面。 首先是对高维数据的非质子化方法的研究。 主要的重点是使用浓度不平等和其他非反应工具对高维数据的协方差和精确矩阵估计。次要重点是此类估计器对实际推论问题的适用性。 第二个方面是对功能数据的对数孔隙随机过程的研究。 主要目标是仔细构建具有足够良好属性的随机过程的非参数类别-e.g。强大的集中行为 - 用于推断功能数据问题，包括语音分析和神经影像学。 最后一个主题是推断变换不变分布。 一个兴趣的问题是确定一种检测数据不变性到某种转换的测试方法。 例如，多元高斯分布具有椭圆形对称性。 第二个问题是在应用这种转换时研究随机变量的浓度和收缩特性。 这是Rademacher平均值的扩展，该扩展具有引人入胜且非常有用的特性。 这项研究建议的影响可能已经达到很远。首先，它旨在开发方法，该方法将广泛适用于一些最复杂和最难以理解的数据，可用于科学研究人员。这包括高维数据，例如基因表达和流行病学研究。它还包括无限尺寸或功能数据，例如神经影像学和其他医学成像。此外，它可以应用于各种类型的空间时间数据，例如气候测量的集合。 除了这样的应用之外，这项研究还将与概率理论和功能分析的主题有密切的联系，从而导致许多统计方法论和统计理论的顶级期刊中引用了许多引用的论文。