Exact and Asymptotic Distribution Theory for General Gaussian Processes

本项目将进一步发展高斯过程及其二次型的精确和渐近分布理论。虽然数据科学的现代进步很大程度上归功于计算方法和计算机技术的快速发展，但统计学和应用概率论的例子比比皆是，在这些例子中，仔细的数学分析可以发现任何计算能力都无法揭示的发现。这个项目就是这样一个例子，它将使用PI在Yule所谓的“无意义”相关性上的工作，这是一个有90年历史的开放问题，去年通过数学分析工具得到了解决。这种明确的计算显示了两个独立连续序列数据之间明显相关性的精确尺度，例如人们在经济学、气候科学、金融和许多其他领域遇到的数据。这种对一个明显的统计悖论的数学解释，将有助于研究数理统计中的其他重要问题。该项目将调查一些重要的开放性问题与概率论中一系列工具之间的可能联系，这些工具的威力数学统计学家才刚刚开始研究。该项目将为莱斯大学和密歇根州立大学的统计学研究生培训提供肥沃的土壤；学生将受益于广泛的机会，从对数学工具的严格研究，到在统计学中的应用，再到在具有重大社会价值的领域的应用。这个项目将研究两个独立或相关高斯过程之间的皮尔逊相关的概率律。第二维纳混沌（正态的二次形式）的分布分析是一套新的工具。这些工具足够灵活，可以通过所谓的Karhunen-Loeve展开来处理任何高斯过程。在应用方面，最引人注目的是，基于这些预测研究的任何统计估计或测试都只需要一个或两个观察结果；这在一些情况下特别有用，例如在环境统计或经济学中，在这些情况下，无法设计实验，并且必须使用及时动态收集的可用可观察数据。本研究的第二个重点是Polya频率函数和相关密度，由于实现了密度可以在第二次维纳混沌中明确表示和扩展，因此使用了一些相同的数学工具。该项目试图证明密度何时是强对数凹的（例如，它的对数有一个远离零的二阶导数）。这个问题，在数理统计中更广泛地用Polya频率函数来表达，具有独立和非同分布指数的和的分布，扩展到一般的二阶混沌分布的情况。该项目可能对统计实践产生重要影响，特别是在比较非平凡时间序列是一项挑战的领域，以及在许多由对数凹性和强对数凹性性质决定的科学领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will further the development of exact and asymptotic distribution theory for Gaussian processes and their quadratic forms. While modern advances in data science owe much progress to computational methods and the rapid growth in computer technology, statistics and applied probability are rife with examples where a careful mathematical analysis allows discoveries that no amount of computational power can uncover. This project is one such example and will use the PI's work on Yule's so called "nonsense" correlation, a 90-year old open problem that was solved last year via mathematical analysis tools. This explicit calculation showed the precise scale of the apparent correlation between two independent continuous series of data, such as what one encounters in economics, climate science, finance, and many other fields. This mathematical explanation of an apparent statistical paradox will enable the investigation of other important questions in mathematical statistics. The project will investigate a possible connection between some important open questions and a set of tools in probability theory whose power mathematical statisticians have only begun to investigate. The project will provide fertile ground for statistics graduate student training at Rice and Michigan State Universities; students will benefit from a wide scope of opportunities, from rigorous study of mathematical tools, to their use in statistics, to applications in fields of great societal value.This project will investigate the probability law of the Pearson correlation between two independent or dependent Gaussian processes. Analyses of distributions in the second Wiener chaos (quadratic forms of normals) are a new set of tools that will be brought to bear. Those tools are flexible enough to handle any Gaussian process via their so-called Karhunen-Loeve expansions. In terms of applications, what is most striking is that any statistical estimation or test based on these projected studies would only require a single or a pair of observations; this is particularly useful for situations, such as in environmental statistics or in economics, where experiments cannot be designed, and one has to work with the available observable data collected dynamically in time. The second emphasis in this study, on Polya frequency functions and related densities, uses some of the same mathematical tools, thanks to a realization that the densities can be represented and expanded explicitly in the second Wiener chaos. The project seeks to prove when a density is strongly log-concave (e.g. its logarithm has a second derivative which is bounded away from zero.) This question, which in mathematical statistics is phrased more broadly in terms of Polya frequency functions, has distribution of sums of independent and non-identically distributed exponentials, expands to the case of general second-chaos distributions. The project could have important consequences in the practice of statistics, especially in areas where comparing non-trivial time series is a challenge, and in many scientific fields informed by properties of log-concavity and strong log-concavity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.