Asymptotic Inference for Locally Stationary Processes

局部平稳过程的渐近推理

• 批准号: 1106460
• 负责人: Michael Nussbaum
• 金额: $ 36.9万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106460&HistoricalAwards=false
• 关键词: Asymptotic; Inference; Locally; Stationary; Processes

Stationarity is a crucial assumption in classical time series analysis. The theory of oscillatory spectra (Priestley, 1965) represents an attempt to overcome the resulting limitations for modeling nonstationary time series. However, it is the more flexible concept of locally stationary processes (Dahlhaus, 1993) which, by extending the theory of oscillatory spectra, provides a suitable framework for a general asymptotic theory of nonstationary processes. A fundamental characteristic of these processes is the time varying spectral density. While the literature on these models is already well developed, several questions of asymptotic inference remain open. Among them, two seem to stand out as most interesting: the possible asymptotic equivalence to a Gaussian white noise model, and the question of optimal exponential rates of large deviation type in testing and estimation problems. Since Le Cam developed the comparison of statistical experiments via their risk functions, many statistical models have been proved to be locally asymptotically normal, with the aim of establishing benchmarks for optimal procedures. For a parametric model of a time varying spectral density, local asymptotic normality has been established in the literature. However, for a better conceptual understanding of asymptotic inference, it is of interest to study the stronger property of asymptotic equivalence to a Gaussian white noise model, valid globally and over nonparametric function classes. As regards large deviation theory for locally stationary processes, some fragments are already available in the literature. A more fully developed theory can be envisaged, yielding not only testing results such as Stein's lemma and the Chernoff bound as special cases, but possibly also insights into the information geometry of these models based on the asymptotic Kullback-Leibler information.Practitioners of statistics and data analysis very often assume that data show more or less similar behavior over different time periods, even when there is clear evidence to the contrary. For instance, this phenomenon can be observed in records of atmospheric turbulence, seismic signals from earthquakes, or speech signals analyzed in biological research. There is a need to develop more refined statistical methods for these cases. An ingenious theoretical solution to this problem has been proposed in the literature, based on the assumption that if data change over time, they often do not do so abruptly, but in a smooth way. This phenomenon is called local stationarity. If the series exhibits these "smooth changes", existing statistical methods can be adapted to smoothly change over time as well, considerably extending the scope of data analysis. The current proposal aims at a more thorough mathematical-statistical investigation of these locally stationary models. It also has a major educational component, as it is intended to accompany the collaboration between the principal investigator and a promising young scientist who has previously been supported with full tuition and stipend from the government of Mexico.

平稳性是经典时间序列分析中的一个关键假设。振荡谱理论（Priestley，1965）代表了一种克服非平稳时间序列建模所产生的局限性的尝试。然而，局部平稳过程的更灵活的概念（Dahlhaus，1993）通过扩展振荡谱理论，为非平稳过程的一般渐近理论提供了合适的框架。这些过程的基本特征是随时间变化的谱密度。尽管有关这些模型的文献已经很成熟，但渐近推理的几个问题仍然悬而未决。其中，有两个似乎最有趣：高斯白噪声模型的可能渐近等价，以及测试和估计问题中大偏差类型的最优指数率问题。自从 Le Cam 通过风险函数开发了统计实验的比较以来，许多统计模型已被证明是局部渐近正态的，目的是为最优程序建立基准。对于时变谱密度的参数模型，文献中已经建立了局部渐近正态性。然而，为了更好地概念性地理解渐近推理，研究渐近等价于高斯白噪声模型的更强的性质是有意义的，该性质在全局和非参数函数类上有效。关于局部平稳过程的大偏差理论，文献中已经有一些片段。可以设想一个更全面发展的理论，不仅产生诸如斯坦因引理和切尔诺夫界限等特殊情况的测试结果，而且还可能基于渐近库尔贝克-莱布勒信息深入了解这些模型的信息几何。统计和数据分析的从业者经常假设数据在不同时间段内表现出或多或少相似的行为，即使有明确的相反证据。例如，这种现象可以在大气湍流的记录、地震的地震信号或生物研究中分析的语音信号中观察到。有必要针对这些情况开发更精细的统计方法。文献中针对这个问题提出了一种巧妙的理论解决方案，它基于这样的假设：如果数据随着时间的推移而变化，它们通常不会突然变化，而是以平稳的方式变化。这种现象称为局部平稳性。如果该系列表现出这些“平滑变化”，则现有的统计方法也可以适应随着时间的推移平滑变化，从而大大扩展数据分析的范围。当前的提案旨在对这些局部平稳模型进行更彻底的数学统计研究。它还具有重要的教育组成部分，因为它的目的是伴随首席研究员和一位有前途的年轻科学家之间的合作，这位年轻科学家此前曾获得墨西哥政府的全额学费和津贴支持。