Statistical Inference for Functional and High-Dimensional Time Series

该研究为功能和高维时间序列的统计推断提供了新的方法，重点是多样本功能数据和面板数据。因此，研究人员扩展了功能数据分析的范围，超出了目前可用的模型。这是相关的，因为所提出的模型在经济学和金融学、地球物理学和工程学中有许多重要的应用。许多函数式数据应用程序的一个共同特征是存在依赖性。然而，到目前为止，功能数据的理论主要集中在独立过程上，功能自回归和线性过程给出了显著的例外。本提案代表了一个全面的研究计划，以开发一个扩展的工具包，用于分析功能和高维时间序列数据。它包含以下部分：开发功能齐全的独立性和平稳性测试，以及不需要降维的诊断测试；考虑到在零假设和备选假设下应该使用不同的算子，并为主成分数量增加的情况提供了新的理论，从而推进了功能主成分分析；介绍和发展方差的功能分析概念的理论基础，包括将功能时间序列观测聚类成组的程序；将面板数据的方法提高到更一般的模型，明确允许观测的高维性，但明显不要求面板的平稳性；并将函数序列与高维时间序列方法相结合，开辟了新的领域。这需要发展复杂的新统计方法，包括（向量值）希尔伯特空间值观测理论的细化和扩展。这项研究还包括一个重要的创新计算组件。为了帮助传播结果，我们计划通过互联网免费提供相关软件。该提案的完成为统计学家和从业者提供了分析不同形式功能数据的新工具。该提案本质上是跨学科的，其应用领域广泛，从金融和经济学（逐点交易数据、几个经济指标的联合运动、政策变化对经济进程的影响）、环境科学（监测空气污染、温度变化、某些极端气象事件的发生变化）到地球物理学（磁力计的磁场读数）。在金融数据的背景下，独立性和平稳性检验可以用来确定，例如，功能自回归模型是否适合于高分辨率资产价格数据。如果是这样，那么进一步的估计技术可以应用于预测资产价值以及经济预测中的其他技术。通过将方差的功能分析应用于从几个不同位置进行的磁场测量，可以根据它们表现出的磁场行为对这些位置进行分类。这可能会影响这些地区无线电通信的实施。因此，这项研究对实践者来说是直接感兴趣的，并将进一步将统计学和科学领域与重要的统计组成部分联系起来。它还推动了数理统计理论的发展。拟议的研究将培养在统计学和应用领域都精通理论和实践的博士生，其中包括女性和少数民族学生。本科生在本研究中的训练和参与也包括在常规课程、独立学习和项目中。

The proposed research provides novel methodology for the statistical inference of functional and high-dimensional time series with a focus on multi-sample functional data and panel data. The investigators thereby extend the range of functional data analysis beyond the currently available models. This is relevant because the proposed models have a number of important applications in economics and finance, geophysics and engineering. A characteristic common to many functional data applications is the presence of dependence. The theory of functional data, however, is as of today mainly focused on independent processes with notable exceptions being given by the functional autoregressive and linear processes. This proposal represents a comprehensive research plan for developing an extended tool-kit for the analysis of functional and high-dimensional time series data. It contains the following parts:development of fully functional tests for independence and stationarity, and diagnostic tests that do not require dimension reduction; advancement of functional principal component analysis by taking into account that different operators should be used under the null and alternative hypothesis and by providing novel theory for the case of an increasing number of principal components; introduction and development of the theoretical foundation of the concept of functional analysis of variance, including procedures to cluster functional time series observations into groups; advancement of the methodology of panel data to more general models, explicitly allowing for the high-dimensionality of the observations but notably not requiring stationarity of the panels; and the breaking of new ground by combining functional series with high-dimensional time series methodology. This requires the development of sophisticated new statistical methodology, including the refinement and extension of the theory of (vector-valued) Hilbert space-valued observations. The research also includes a significant innovative computational component. To aid the dissemination of results, we plan to make the relevant software freely available via the Internet. Completion of the proposal gives statisticians and practitioners new tools for analyzing different forms of functional data.The proposal is interdisciplinary in nature, with applications in diverse fields ranging from finance and economics (tick-by-tick transaction data, joint movement of several economic indicators, the effect of policy changes on economic processes), environmental science (monitoring air pollution, changes in temperature, change in the occurrences of certain meteorological extremes), and to geophysics (magnetic field readings of magnetometers). In the context of financial data, independence and stationarity testing can be used to determine if, for example, the functional autoregressive model is appropriate for high resolution asset price data. If so then further estimation techniques can be applied towards predicting asset values as well as other techniques in economic forecasting. By applying the functional analysis of variance to magnetic field measurements taken from several different locations one may categorize these locations according to the magnetic field behavior they exhibit. This may influence the implementation of radio communication in these areas. The research is therefore of immediate interest for practitioners and will further connect statistics and fields of science with a significant statistical component. It also advances the theory of mathematical statistics. The proposed research produces doctoral students, among them female and minority students, theoretically and practically versed in both statistics and an area of application. The training and involvement of undergraduate students in this research is also included through regular coursework, independent study and projects.