Higher order accuracy of bootstrap methods for temporal and spatial processes

该项目的重点是研究时间序列和空间数据的常见重采样方法的高阶性质，并开发新的推断方法，以提高现有方法的精度和稳定性。具体地说，这个项目集中于(I)发展相关性下的学生化统计的Edgeworth展开理论，(Ii)研究时间序列数据的Bootstrap近似的高阶精度，(Iii)发展旨在获得更高阶精度的重采样方法，(Iv)经验地选择最佳块长度的非参数方法，(V)产生更稳定的总体参数估计的Bootstrap的汇集方法，(Vi)研究Bootstrap近似在空间预测问题中的精度，研究不规则空间数据点的Bootstrap近似方法的精度。在天文学、大气科学、经济学、地质学、水文学、物理学等许多科学领域都存在着时间和空间相关性的数据，用现有的统计方法分析这些数据集面临一些局限性。这主要是因为现有的统计方法大多依赖于强有力的结构性(即参数模型)假设，而这些假设往往不足以捕捉数据的所有重要特征。该项目力求(1)开发新的方法(基于已知的重采样方法)，在没有强有力的结构性假设的情况下提供对不确定性的有效评估；(2)开发理论工具，以研究各种重采样方法对与时间和空间相关的数据的最优性质。

The emphasis of the project is on investigating higher order properties of common resampling methods for time-series and spatial data and on development of new inference methods that improve the accuracy and stability of existing methods. Specifically, this project concentrates on (i) developing Edgeworth expansion theory for Studentized statistics under dependence, (ii) investigating higher order accuracy of bootstrap approximations for time series data, (iii) developing resampling methods with an aim towards achieving higher order accuracy, (iv) a nonparametric method for the selection of optimal block length empirically,(v) a pooling method for the bootstrap that yields more stable estimators of population parameters, (vi) investigating accuracy of bootstrap approximation in spatial prediction problems, and(vii) investigating accuracy of bootstrap approximation for spatial data for irregularly spaced data-points.Data exhibiting temporal and spatial dependence appear in many areas of sciences, such as Astronomy, Atmospheric Sciences, Economics, Geology, Hydrology, Physics, etc. Analyses of such data sets using current statistical methodology face some limitations. This is primarily due to the fact that the existing statistical methodology mostly relies on strong structural (i.e., parametric model) assumptions that are often inadequate to capture all important features of the data. This project seeks to(i) develop new methodology (based on what are known as Resampling Methods) that provides valid assessment of uncertainty without strong structural assumptions and (ii) develop theoretical tools to investigate optimality properties of various resampling methods for time- and space-dependent data.