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

正确的模型规范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，因此现在可以将时间序列分析为非高斯或非线性。因此，需要有可靠的测试来确定线性或非线性，并有全面的工具来进行适当的分析。对现有的基于双谱的时间序列高斯性和线性测试的初步调查揭示了许多有待研究的领域。对于各种类型的非线性，将进行双谱和归一化双谱的不同函数的检验。第二个研究领域是开发一种检验统计量，当在估计光谱量时使用不同的平滑程序时，这种统计量不会产生很大不同的值。主要研究者希望在进行这些检查之后，将开发一个检验统计量，不仅可靠地检验线性，而且在拒绝零假设时有助于诊断非线性类型。本研究的第二个目标是继续在S-Plus中开发时间序列分析包，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。由于封装是S-Plus，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，包和文档将提交给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.