Mathematical Sciences: Developing a Bispectral-Based Test for Gaussianity and Linearity of a Time Series

正确的模型规格可以说是可靠的时间序列分析的最关键的方面。 在过去的十年中，已经取得了重大进展，因此现在可以分析非高斯或非线性的时间序列。 因此，期望具有用于确定线性或非线性的可靠测试，并且具有用于执行适当分析的综合工具。 对现有的基于双谱的时间序列的高斯性和线性检验的初步调查揭示了许多研究领域。 将对各种类型的非线性的双谱和归一化双谱的不同函数进行检查。 调查的第二个领域是一个测试统计量的发展，将不会导致在不同的平滑过程中使用估计光谱量时，在很大程度上不同的值。 主要研究者希望在进行此类检查后，将开发一种检验统计量，不仅可以可靠地检验线性，而且可以在拒绝零假设时帮助诊断非线性的类型。 本研究的第二个目标是继续开发一个时间序列分析软件包在S-Plus，其中将包括用于分析非线性时间序列的设备。 该软件包将具有可用于线性和非线性时间序列分析的功能。 由于该软件包是在S-Plus中，相关时间序列数量的图形检查是一个“内置功能”。" 还将编制文件。 完成后，这个包连同文档沿着将提交给statlib供公众使用。

Correct model specification is arguably the most critical aspect of reliable time series analysis. In the past decade, significant advances have been made so it is now possible to analyze a time series as non-gaussian or non-linear. Therefore, it is desirable to have reliable tests for determining linearity or non-linearity and to have comprehensive tools for performing proper analysis. An initial investigation of existing bispectral-based tests for gaussianity and linearity of a time series revealed many areas for study. Examinations of different functions of the bispectrum and normalized bispectrum for various types of non-linearity will be conducted. A second area of investigation is the development of a test statistic that will not result in vastly different values when different smoothing procedures are used in estimating spectral quantities. It is the hope of the principal investigator that after such examinations are conducted, a test statistic will be developed that not only reliably tests for linearity, but will aid in diagnosing the type of non-linearity when rejecting the null hypothesis. A second goal of this research is to continue to develop a time series analysis package in S-Plus which will include devices for analyzing non-linear time series. The package will have functions that may be used for linear and non-linear time series analysis. Since the package is in S-Plus, graphical examination of relevant time series quantities is a "built-in feature." Documentation will also be developed. When complete, the package, along with documentation, will be submitted to statlib for public use.