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中，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。这个研究项目有两个主要目标。首先，它寻求开发新的数学工具来分析最近一种复杂的相关数据统计方法，这种方法在实证研究中产生了惊人的准确结果，但缺乏任何理论依据。希望新的理论工具将导致进一步完善现有的相关数据统计方法。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，例如从数百名患者的数据中确定数千个基因集合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的模型假设的情况下解决重要的推理问题。该项目旨在开发一些关键的理论工具和非参数统计方法来分析时间序列和高维数据。具体而言，本项目将侧重于(i)开发时间序列中“固定-b”渐近方法的渐近展开式结果，该结果在一些实证研究中显示出比传统方法有显着改进，但理论基础很少；（ii）研究一些一般类别的统计检验（例如Wald检验）和一些最近提出的非标准经验似然检验的高阶性质，两者都是在“固定-b”公式下；（iii）在时间序列中开发新的块自举关键量，使其精度接近于独立状态下的自举；（iv）利用近似理论和巴拿赫空间理论的一些新工具，得到了稀疏度下高维的渐近展开结果；(v)应用（iv）的渐近展开结果研究高维统计方法的渐近性质中的“相变”现象，以及（vi）研究高维后变量选择推理的重采样方法的性质。

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.