Limit Theorems and Statistical Inference for Ergodic Processes

遍历过程的极限定理和统计推断

• 批准号: 0102268
• 负责人: Michael Woodroofe
• 金额: $ 24万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102268&HistoricalAwards=false
• 关键词: Limit; Theorems; Statistical; Inference; Ergodic

Abstract 0102268Limit Theorems and Statistical Inference for Ergodic ProcessesA major goal of the project is to develop a new approach to the change point problem in which the abrupt change of the latter is replaced by an arbitrary monotonic change. The new procedure uses a penalized likelihood ratio statistic for testing equality of means against a non-decreasing trend, derived for independent normal observation errors. The properties of the test can be studied in the more general context of dependent, but stationary and ergodic errors. Applications of such procedures should be evident in the analysis of climate changes results from cataclysmic events or legal intervention, such as the required reduction on vehicle emissions.Current work by the principal investigator and students has determined the asymptotic null distribution of the test statistic for stationary ergodic errors under modest conditions, thus allowing application to historical data sets, like weather data. Remaining questions include developing a sequential analogue for applications to quality control, alternative penalizations, and estimating a variance parameter after an isotonic regression. A second major goal of the project is to develop asymptotic distribution theory in a context that is applicable to the first. The central limit theorem will be studied for additive functionals of a Markov chain with special attention to chains in which the current state is a function of the previous state and an independent variable. Many linear and non-linear time series models are of this form. Conditions for the existence of a stationary distribution have been widely studied for such processes, but there is much less work on central limit theory for their additive functionals. The principal investigator plans to develop central limit theory in this context. Previous work has shown that additive functionals can be written as a martingale plus a remainder term of smaller order in many cases, and then asymptotic normality can be deduced from the martingale central limit theorem. This approach does not require Harris recurrence or other strong forms of asymptotic independence. It will be developed, and statistical applications explored, especially applications to the modified change point problem. Other statistical applications include setting approximate confidence intervals. In some cases, approximate confidence intervals may be obtained from a multivariate central limit theorem. For others, it is necessary to develop tightness of empirical processes, and this question will be studied. In highly structured models, it is possible to go beyond asymptotic normality to (Edgeworth like) asymptotic expansions from which corrected confidence intervals can be formed, intervals whose actual coverage probability converges to the nominal value at a fast rate. A third major objective of project is to develop such expansions. Previous work by the principal investigator, co-workers, and students has developed expansions of this nature for adaptively designed linear models and auto regressive processes. This work will be extended to processes whose finite dimensional distributions form exponential families, a large class of processes that includes Markov Chains and many semi-Markov processes.

摘要0102268遍历过程的极限定理和统计推断该项目的主要目标是开发一种新的方法来解决变点问题，其中后者的突变被任意单调变化所取代。 新的程序使用惩罚似然比统计量来测试非递减趋势的平均值的相等性，该方法是针对独立的正态观测误差推导的。 检验的性质可以在更一般的相关、平稳和遍历误差的情况下进行研究。 这些程序的应用在分析气候变化的结果，从灾难性的事件或法律的干预，如所需的减少车辆排放量，应是显而易见的。目前的工作，由主要研究员和学生已确定的渐近零分布的平稳遍历错误的测试统计量在温和的条件下，从而允许应用到历史数据集，如天气数据。 剩下的问题包括开发一个连续的类似物应用到质量控制，替代惩罚，估计方差参数后，保序回归。 该项目的第二个主要目标是在适用于第一个目标的背景下开发渐进分布理论。 中心极限定理将研究添加剂泛函的马尔可夫链，特别注意链中的当前状态是一个函数的前一个状态和一个独立的变量。 许多线性和非线性时间序列模型都是这种形式。 对于这类过程，平稳分布存在的条件已经得到了广泛的研究，但是对于它们的可加泛函的中心极限理论的研究却少得多。 主要研究者计划在此背景下发展中心极限理论。 以往的工作表明，在许多情况下，可加泛函可以写成一个鞅加上一个低阶的余项，然后可以从鞅中心极限定理推导出渐近正态性。 这种方法不需要Harris递归或其他强形式的渐近独立性。 它将被开发，并探讨统计应用，特别是应用到修改的变点问题。其他统计应用包括设置近似置信区间。 在某些情况下，近似置信区间可以从多元中心极限定理获得。 对于其他人来说，有必要发展经验过程的紧密性，这个问题将被研究。 在高度结构化的模型中，有可能超越渐近正态性（Edgeworth样）渐近展开，从中可以形成校正的置信区间，其实际覆盖概率以快速的速度收敛到标称值的区间。 该项目的第三个主要目标是开发这种扩展。 主要研究者、同事和学生以前的工作已经为自适应设计的线性模型和自回归过程开发了这种性质的扩展。 这项工作将扩展到过程的有限维分布形成指数族，一个大类的过程，包括马尔可夫链和许多半马尔可夫过程。