The Weighted Bootstrap and Berry-Esseen Bounds in High Dimensions

重采样方法在许多应用中被广泛用于统计推断。特别是，众所周知，引导在可用数据量相当小的情况下执行得很好。在许多现代应用中，数据是复杂和高维的，这推动了统计推断中新方法的发展，包括重新抽样方法。在这个项目中，研究人员将在高维框架中研究具有有限数据量的各种统计模型的加权Bootstrap过程。本研究的主要目的是了解加权Bootstrap的基本性质，如其局限性和准确性，并提出高维环境下的重采样方法。在这个项目中，调查者将研究高维非渐近框架下样本平均的函数分布的逼近问题，对于各种函数类。我们将研究两种彼此密切相关的基本类型的近似：使用加权Bootstrap过程的近似和Berry-Esseen型不等式。在这两种情况下，研究人员的目标都是建立高阶近似界，以扩展经典的高斯近似理论，并在维度和样本大小方面产生相当大的精度改进。研究的重点是结果边界的最佳性，以及误差项的显式形式。本项目研究的另一个重要方向是将所提出的高阶近似方法推广到重尾分布的情况，这在金融和工程中的许多应用中都起着重要的作用。该项目的工作旨在应用和进一步发展各种现代技术，如高阶逼近和比较不等式，多线性对称形式的集中不等式，高维高斯测度的几何性质，以及与多元矩问题相关的方法。

Resampling methods are widely used for statistical inference in numerous applications. In particular, bootstrapping is known to perform well in situations when the amount of available data is rather small. In many modern applications the data are complex and high-dimensional, motivating development of new approaches in statistical inference, including resampling methods. In this project, the investigator will study weighted bootstrap procedures in a high-dimensional framework with a limited amount of data, for various classes of statistical models. The main goals of this research are to understand essential properties of the weighted bootstrap, such as its limitations and accuracy, and to advance resampling methods in high-dimensional settings.In this project, the investigator will study the problem of approximation in distribution of a function of a sample average in a high-dimensional non-asymptotic framework, for various classes of functions. Two basic types of approximations, which are closely related to each other, will be studied: an approximation using the weighted bootstrap procedure, and Berry-Esseen type inequalities. In both cases, the investigator aims to establish higher-order approximation bounds that extend classical Gaussian approximation theory and yield considerable improvements in accuracy with respect to both dimension and sample size. The study will be focused on optimality of the resulting bounds, and on explicit form of the error terms. Another important direction of the research in this project is extension of the proposed higher-order approximation methodology to the case of heavy-tailed distributions, which play an important role in many applications in finance and engineering. The work on the project aims to employ and further develop various modern techniques, such as higher-order approximation and comparison inequalities, concentration inequalities for multilinear symmetric forms, geometric properties of Gaussian measures in high dimensions, and methods related to multivariate moment problems.