The Weighted Bootstrap and Berry-Esseen Bounds in High Dimensions

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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.