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

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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.