Discriminant analysis of time series

This thesis is concerned with determining whether two time series have been generated by underlying processes with the same spectral shape. This question has arisen in the literature in a variety of disciplines, including signal processing, economics and the natural sciences. Most of the methods that have been proposed to discriminate between spectral densities are nonparametric and are based on the periodogram. The statistical properties of these tests, however, are generally developed on the assumption that the underlying processes are Gaussian white noise. One such method considers the range of the logarithm of the ratios of periodograms at the Fourier frequencies. The asymptotic distribution of this range statistic is derived in the Gaussian white noise case, and it is conjectured that the result holds in the non-Gaussian and coloured case. This conjecture is supported by a simulation study. A parametric approach to spectral discrimination involves fitting autoregressions to the time series and then testing for differences in spectral shape using a likelihood ratio procedure. This approach is reviewed in detail, and algorithms are presented for carrying out the test. The statistical properties of the test are examined using simulations. A power study shows that the parametric approach has a higher empirical power than nonparametric tests. Future work is discussed, which includes extending the parametric approach to a more general class of models.