Exact and Asymptotic Distribution Theory for General Gaussian Processes

一般高斯过程的精确渐近分布理论

基本信息

  • 批准号:
    1811779
  • 负责人:
  • 金额:
    $ 25万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2018
  • 资助国家:
    美国
  • 起止时间:
    2018-07-01 至 2022-06-30
  • 项目状态:
    已结题

项目摘要

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.
这个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充满了这样的例子,即仔细的数学分析可以发现任何计算能力都无法发现的发现。这个项目就是一个这样的例子,它将使用PI对Yule所谓的“无意义”相关性的研究,这是一个90年前的开放问题,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的连续数据系列之间的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使数理统计中其他重要问题的研究成为可能。该项目将调查一些重要的开放性问题和概率论中的一套工具之间可能存在的联系,这些工具的威力数学统计学家才刚刚开始调查。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,从严格的数学工具学习,到它们在统计学中的应用,再到具有巨大社会价值的领域的应用。在第二个维纳混沌(二次形式的法线)的分布分析是一套新的工具,将承担。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只需要一个或一对观察结果;这对于环境统计或经济学等情况特别有用,因为实验无法设计,并且必须使用动态收集的可观察数据。在这项研究中的第二个重点,波利亚频率函数和相关的密度,使用一些相同的数学工具,由于实现的密度可以表示和扩展明确的第二维纳混沌。该项目旨在证明密度何时是强对数凹的(例如,其对数具有远离零的二阶导数)。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之和的分布,扩展到一般二次混沌分布的情况。该项目可能会在统计实践中产生重要影响,特别是在比较非平凡时间序列是一项挑战的领域,以及在许多由对数周期和强对数周期特性所告知的科学领域。该奖项反映了NSF的法定使命,并被认为值得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来支持。

项目成果

期刊论文数量(26)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Extreme-strike asymptotics for general Gaussian stochastic volatility models
一般高斯随机波动率模型的极端走向渐近
  • DOI:
    10.1007/s10436-018-0338-z
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    1
  • 作者:
    Gulisashvili, Archil;Viens, Frederi;Zhang, Xin
  • 通讯作者:
    Zhang, Xin
Fiscal stimulus as an optimal control problem
  • DOI:
    10.1016/j.spa.2021.05.009
  • 发表时间:
    2021-06
  • 期刊:
  • 影响因子:
    1.4
  • 作者:
    Philip A. Ernst;Michael B. Imerman;L. Shepp;Quan Zhou
  • 通讯作者:
    Philip A. Ernst;Michael B. Imerman;L. Shepp;Quan Zhou
A martingale approach for fractional Brownian motions and related path dependent PDEs
  • DOI:
    10.1214/19-aap1486
  • 发表时间:
    2017-12
  • 期刊:
  • 影响因子:
    0
  • 作者:
    F. Viens;Jianfeng Zhang
  • 通讯作者:
    F. Viens;Jianfeng Zhang
A probabilistic approach to Adomian polynomials
  • DOI:
    10.1080/07362994.2020.1755312
  • 发表时间:
    2020-05
  • 期刊:
  • 影响因子:
    1.3
  • 作者:
    P. Vellaisamy;F. Viens
  • 通讯作者:
    P. Vellaisamy;F. Viens
Stability and busy periods in a multiclass queue with state-dependent arrival rates
具有与状态相关的到达率的多舱位队列中的稳定性和繁忙期
  • DOI:
    10.1007/s11134-018-9587-9
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    1.2
  • 作者:
    Ernst, Philip A.;Asmussen, Søren;Hasenbein, John J.
  • 通讯作者:
    Hasenbein, John J.
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Philip Ernst其他文献

The Minimax Wiener Sequential Testing Problem
极小极大维纳序贯检验问题
  • DOI:
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Philip Ernst;Hongwei Mei
  • 通讯作者:
    Hongwei Mei
Touching is not taboo
  • DOI:
    10.1016/s0197-4572(80)80072-3
  • 发表时间:
    1980-09-01
  • 期刊:
  • 影响因子:
  • 作者:
    Philip Ernst;Jeanne Shaw
  • 通讯作者:
    Jeanne Shaw

Philip Ernst的其他文献

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{{ truncateString('Philip Ernst', 18)}}的其他基金

A Symposium on Optimal Stopping
最佳停止研讨会
  • 批准号:
    1822487
  • 财政年份:
    2018
  • 资助金额:
    $ 25万
  • 项目类别:
    Standard Grant

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    EP/Y029089/1
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    2024
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Conference: Geometric and Asymptotic Group Theory with Applications 2024
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    2024
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Asymptotic patterns and singular limits in nonlinear evolution problems
非线性演化问题中的渐近模式和奇异极限
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