Higher Order Asymptotics for Some Nonstandard Problems in Time Series and in High Dimensions

相关和高维数据经常出现在许多科学领域，包括大气科学，金融和分子遗传学，以及越来越多的日常活动，如社交网络。虽然正在生成大量数据并可用于分析，但传统方法往往无法在此类应用中获取信息。该研究项目有两个主要目标。 首先，它试图开发新的数学工具，用于分析最近复杂的相关数据统计方法，这些方法在实证研究中产生了非常准确的结果，但缺乏任何理论依据。人们希望，新的理论工具将导致进一步完善现有的相关数据的统计方法。该项目的第二部分涉及高维数据的复杂推理问题，其中未知参数的数量远远超过样本量，例如从几百名患者的数据中确定几千个基因中的几个重要基因的作用。该项目旨在开发理论和方法统计工具，使研究人员能够在没有严格模型假设的情况下解决重要的推理问题。该项目旨在为时间序列和高维数据的分析开发一些关键的理论工具和非参数统计方法。具体而言，本项目将侧重于（i）为时间序列中的“固定b”渐近方法开发渐近展开结果，该方法在几项实证研究中显示出对传统方法的显着改进，但几乎没有理论基础;（ii）调查一些一般类别的统计检验的高阶性质（例如，Wald检验）和一些最近提出的非标准经验似然检验，两者都在“固定b”公式下;（iii）为时间序列中的块引导开发新的关键量，几乎匹配独立下的引导精度;（iv）通过利用近似理论和Banach空间理论中的一些新工具，在稀疏下开发高维渐近展开结果;（v）应用（iv）的渐近展开结果研究高维统计方法渐近性质中的“相变”现象;（vi）研究高维后变量选择推理的回归方法的性质。

Correlated and high dimensional data appear routinely in many areas of sciences, including atmospheric sciences, finance, and molecular genetics, as well as in an ever increasing number of everyday activities such as social networking. While a vast amount of data are being generated and are available for analyses, traditional methods often fail to elicit information in such applications. This research project has two major goals. First, it seeks to develop new mathematical tools for analyzing a recent complex statistical approach for correlated data that has been known to produce astonishingly accurate results in empirical studies, but lacks any theoretical justification. It is hoped that the new theoretical tool will lead to further refinements of existing statistical methodology for correlated data. The second part of the project is concerned with complex inferential issues for high dimensional data where the number of unknown parameters far exceeds the sample size, such as determining the role of a few important genes among a collection of several thousand genes from data on a few hundred patients. The project seeks to develop theoretical and methodological statistical tools to enable researchers to address important inference questions without stringent model assumptions. The project aims to develop some critical theoretical tools and nonparametric statistical methodology for the analysis of time series and high dimensional data. Specifically, this project will focus on (i) developing asymptotic expansion results for the "fixed-b" asymptotic approach in time series that has shown significant improvement over traditional methods in several empirical studies but with very little theoretical underpinning; (ii) investigating higher order properties of some general classes of statistical tests (e.g., Wald tests) and of some more recently proposed nonstandard empirical likelihood tests, both under the "fixed-b" formulation; (iii) developing new pivotal quantities for block bootstrap in time series that nearly match the accuracy of bootstrap under independence; (iv) developing asymptotic expansion results in high dimensions under sparsity by exploiting some novel tools from approximation theory and Banach space theory; (v) applying the asymptotic expansion results from (iv) to investigate the "phase transition" phenomenon in asymptotic properties of statistical methods in high dimensions, and (vi) investigating properties of resampling methods for post-variable selection inference in high dimensions.