Asymptotic Inference for Locally Stationary Processes

平稳性是经典时间序列分析中的一个重要假设。振荡谱理论(Priestley，1965)代表了克服非平稳时间序列建模的局限性的一种尝试。然而，更灵活的局部平稳过程的概念(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一个合适的框架。这些过程的一个基本特征是时变的谱密度。虽然关于这些模型的文献已经很好地发展了，但关于渐近推断的几个问题仍然悬而未决。其中，有两个问题似乎最为有趣：高斯白噪声模型的渐近等价性，以及检验和估计问题中大偏差型的最优指数率问题。自从Le Cam开发了通过风险函数进行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分布。然而，为了更好地理解渐近推理的概念，研究高斯白噪声模型的渐近等价的更强性质是有意义的，该模型在非参数函数类上是全局有效的。关于局部平稳过程的大偏差理论，文献中已经有了一些片段。可以设想一个更全面的理论，不仅产生像Stein引理和Chernoff界这样的测试结果作为特例，而且可能基于渐近的Kullback-Leibler信息来洞察这些模型的信息几何。统计和数据分析的实践者经常假设数据在不同的时间段表现出或多或少的相似行为，即使有明显的相反的证据。例如，这种现象可以在大气湍流记录、地震地震信号或生物研究中分析的语音信号中观察到。有必要为这些情况开发更精细的统计方法。文献中已经提出了一种巧妙的理论解决方案，该方案基于这样的假设：如果数据随着时间的推移而改变，它们通常不会突然改变，而是以一种平稳的方式进行。这种现象称为局部平稳性。如果序列表现出这些“平稳变化”，现有的统计方法也可以适应随着时间的变化而平稳变化，大大扩大了数据分析的范围。目前的建议旨在对这些局部固定模型进行更彻底的数学统计调查。它也有一个重要的教育部分，因为它是为了配合首席研究人员和一位有前途的年轻科学家之间的合作，这位年轻科学家以前得到了墨西哥政府的全额学费和津贴支持。

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.