Higher order accuracy of bootstrap methods for temporal and spatial processes

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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.