Non-asymptotic inference for high and infinite dimensional data

在分析高维和无限维数据时，需要超越经典渐近理论的新颖统计技术。其中一类替代方法是精确和排列测试，旨在表征数据的有限样本分布[Pigoli等人，2014，Cabassi等人，2017]。然而，这种方法往往陷入组合枚举的计算复杂性，因此不能完全实现。本研究提案旨在通过仔细构建和实施集中不等式，即所谓的非渐近独立性理论，来解决表征数据有限样本分布的相同问题[Boucheron et al., 2013]。这项研究有三个方面。******首先是研究高维数据的非渐近方法。主要的焦点是使用浓度不等式和其他非渐近工具对高维数据进行协方差和精确矩阵估计。第二个重点是这些估计对实际推理问题的适用性。******第二个方面是对函数数据的对数凹随机过程的研究。主要目标是仔细构造具有足够好的特性的随机过程的非参数类。强集中行为-用于功能数据问题的推理，包括语音分析和神经成像。******最后一个主题是关于变换不变分布的推论。我们感兴趣的一个问题是确定一种测试方法来检测数据对某些转换的不变性。例如，多元高斯分布具有椭球对称。第二个问题是研究应用这种变换时随机变量的集中和收缩性质。这是Rademacher平均的扩展，它产生了迷人的和非常有用的性质。******这个研究计划的影响是相当深远的。它的主要目的是开发一种方法，这种方法将广泛适用于科学研究人员可以获得的一些最复杂和最不可思议的数据。这包括高维数据，如基因表达和流行病学研究。它还包括无限维度或功能数据，如神经成像和其他医学成像。此外，它还可以应用于各种类型的时空数据，如气候测量数据的收集。除了这些应用之外，本研究将与概率论和泛函分析的主题密切相关，从而在统计方法和统计理论的顶级期刊上发表许多被引用的论文。

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.