The Weighted Bootstrap and Berry-Esseen Bounds in High Dimensions

在许多应用中，恢复方法被广泛用于统计推断。特别地，已知自举在可用数据量相当小的情况下表现良好。在许多现代应用中，数据是复杂的和高维的，这促使统计推断中的新方法的发展，包括重新排序方法。在这个项目中，研究者将在一个高维框架中，用有限的数据量，研究各种统计模型的加权自举过程。本研究的主要目的是了解加权自举法的局限性和准确性等基本性质，并推进高维环境下的再分配方法。在本项目中，研究者将研究在高维非渐近框架下，各类函数的样本平均值的函数分布的近似问题。两个基本类型的近似，这是密切相关的，将被研究：近似使用加权自助程序，和贝里-Esseen型不等式。在这两种情况下，研究者的目的是建立高阶近似界，扩展经典高斯近似理论，并产生相当大的改善精度方面的尺寸和样本量。该研究将集中在最优性的结果的界限，并在显式形式的误差项。该项目的另一个重要研究方向是将所提出的高阶近似方法扩展到重尾分布的情况，重尾分布在金融和工程中的许多应用中起着重要作用。该项目的工作旨在采用和进一步发展各种现代技术，如高阶近似和比较不等式，多线性对称形式的浓度不等式，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.