Mathematical Sciences: Nonlinear Modeling in Continuous Time, Delayed Autoregressive Processes, and Chaos

建议：DMS 9504798 PI（s）：Kung-Sik Chan和Osnat Stramer机构：爱荷华州大学标题：连续时间非线性建模，延迟自回归 过程与混沌 摘要： 在分析不规则采样的非线性时间序列数据时，自然会考虑（未观察到的）潜在连续时间过程的非线性连续时间模型。 由于这类非线性模型的似然函数的难以确定性，现有的估计方法在文献中有一定的随意性。这项研究有助于非线性连续时间建模的基础上（可能不规则采样）离散时间观测的统计推断。本研究共分四个部分。 在第一部分中，研究人员研究了一般类非线性连续时间自回归过程的极大似然估计，缩写为NLCAR（p）。 研究人员推导出似然函数的新表达式 这可以通过模拟来评估。 调查人员 实现基于仿真的最大似然估计过程的一些类别的NLCAR（p）模型。 然后，研究人员研究了最大值的抽样特性， 这样得到的似然估计。在第二部分中，调查人员 应用拉格朗日乘数检验来检测 非线性和其他诊断目的，以及 研究其大样本性质。 在第三部分中，调查人员 将遍历性和稳定性相关的方法扩展到连续时间模型。 在第四部分中，研究了状态空间为无限维Hilbert空间的时滞连续时间非线性自回归过程。 他们还研究了递归（平稳性）属性和 延迟非线性自回归的统计推断 流程. 在这项研究中，调查人员研究了新的统计方法， 用于分析在可能不相等的时间间隔上获取的数据。 这种数据被称为不规则采样时间序列，经常出现在不同的领域，包括 商业预测、环境统计、医学统计、物理和工程科学。研究人员研究计算机密集型 提供相关研究工具的方法 和测试 不规则采样时间序列数据的非线性结构。 研究人员还开发了一些数理统计和概率理论的基础，以了解新的方法。 努力程度声明 在建议的支持水平下，PI将尽一切努力， 满足项目的原始范围和工作水平。

Proposal: DMS 9504798 PI(s): Kung-Sik Chan and Osnat Stramer Institution: University of Iowa Title: Nonlinear Modeling in Continous Time, Delayed Autoregressive Processes, and Chaos Abstract: In analyzing nonlinear time series data sampled irregularly, it is natural to consider non-linear continuous-time models for the (unobserved) underlying continuous-time process. Owing to the intractability of the likelihood function for such kind of nonlinear models, the existing estimation methods in the literature are somewhat ad hoc. This research contributes to the statistical inference for non-linear continuous-time modeling based on (possibly irregularly sampled) discrete-time observations. This research consists of four parts. In part I, the investigators study maximum likelihood estimation for the general class of nonlinear continuous-time autoregressive process, abbreviated by NLCAR(p). The investigators derive a new expression for the likelihood function which can be evaluated via simulation. The investigators implement the simulation-based maximum likelihood estimation procedure for some classes of NLCAR(p) models. The investigators then study the sampling properties of the maximum likelihood estimator so obtained. In part II, the investigators apply the Lagrange Multiplier tests for detecting non-linearity and also those for other diagnostic purposes, and study their large sample properties. In part III, the investigators extend an approach relating ergodicity and stability to continuous-time models. In part IV, the investigators study delayed continuous-time non-linear autoregressive processes whose state space is an infinite-dimensional Hilbert space. They also study the recurrence (stationarity) properties and the statistical inference for the delayed non-linear autoregressive processes. In this research, the investigators study new statistical methods useful for analyzing data which are taken over possibly unequal ti me intervals. Such kind of data is known as irregularly sampled time series, and occurs frequently in diverse areas including business forecasting, environmental statistics, medical statistics, physical and engineering science. The investigators study computer-intensive methods which provide the relevant tools for studying and testing for the nonlinear structure of irregularly sampled time series data. The investigators also develop some mathematical statistics and probability theory fundamental to the understanding of the new methods. LEVEL OF EFFORT STATEMENT At the recommended level of support, the PI will make every attempt to meet the original scope and level of effort of the project.