Mathematical Sciences: Computer-Intensive Methods for the Statistical Analysis of Time Series and Random Fields

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The statistical analysis of time series and random fields is vital in many diverse scientific disciplines. The general goal of this project is to develop methods of inference for the analysis of time series and random fields that do not rely on unrealistic or unverifiable model assumptions. Typical inferential methods in the data-dependent setting rest upon strong assumptions. In contrast, bootstrap resampling or computer-intensive methods offer viable approaches to obtaining valid distributional approximations while assuming very little about the stochastic mechanism generating the data. Many important questions need to be addressed in order for these modern approaches to be applied safely in practice. The main issues we wish to tackle include the following: finding general conditions for asymptotic validity of computer-intensive methods, especially subsampling, in the presence of nonstationarity; higher-order comparison of block-resampling and subsampling; developing computationally efficient and accurate estimates of standard error, and the corresponding improvement on confidence regions for parameters of interest; optimal choice of design parameters, such as block size, as well as practical guidelines for implementation; assessing goodness of fit in time series settings; and finally interval estimation with lattice or non-lattice random field data. Addressing these general problems fruitfully will have many practical applications. Applications of statistical methods for time series and spatial data are well-known and numerous, especially in the fields of physics, engineering, acoustics, geostatistics, medicine, econometrics, ecology, forestry, seismology and others. The general purpose of this research proposal is to develop inferential methods for dependent data that does not rely on strong model assumptions. These methods are computer-intensive, very generally applicable, flexible, and offer solutions to problems when there are no alternatives; how ever, further mathematical study of these procedures is needed in order to fully under their potential and limitations.