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

时间序列和随机场的统计分析在许多不同的科学学科中至关重要。 该项目的总体目标是开发用于分析时间序列和随机场的推理方法，这些方法不依赖于不现实或无法验证的模型假设。 典型的推理方法在数据依赖的设置依赖于强有力的假设。 相比之下，自举响应或计算机密集型方法提供了可行的方法来获得有效的分布近似值，同时对生成数据的随机机制假设很少。 为了使这些现代方法在实践中安全应用，需要解决许多重要问题。 我们希望解决的主要问题包括：在非平稳性存在的情况下，找到计算机密集型方法，特别是子采样方法的渐近有效性的一般条件;块恢复和子采样的高阶比较;发展计算效率高和准确的标准误差估计，以及对感兴趣参数的置信区域的相应改进;设计参数的最佳选择，如块大小，以及实施的实际指导方针;评估时间序列设置中的拟合优度;最后是格点或非格点随机场数据的区间估计。 卓有成效地解决这些普遍问题将有许多实际应用。 时间序列和空间数据的统计方法的应用是众所周知的和众多的，特别是在物理学、工程学、声学、地质统计学、医学、计量经济学、生态学、林业、地震学等领域。 本研究提案的总体目的是为不依赖于强模型假设的相关数据开发推理方法。这些方法是计算机密集型的，非常普遍适用的，灵活的，并提供解决问题时，没有替代方案;然而，这些程序需要进一步的数学研究，以充分发挥其潜力和局限性。

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.