A Frequency Domain Test for Propriety of Complex-Valued Vector Time Series

This paper proposes a frequency domain approach to test the hypothesis that a stationary complex-valued vector time series is proper, i.e., for testing whether the vector time series is uncorrelated with its complex conjugate. If the hypothesis is rejected, frequency bands causing the rejection will be identified and might usefully be related to known properties of the physical processes. The test needs the associated spectral matrix that can be estimated by multitaper methods by using, say, <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula> tapers. Standard asymptotic distributions for the test statistic are of no use since they would require <inline-formula> <tex-math notation="LaTeX">$K \rightarrow \infty,$</tex-math></inline-formula> but, as <inline-formula> <tex-math notation="LaTeX">$K$</tex-math></inline-formula> increases so does resolution bandwidth that causes spectral blurring. In many analyses, <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula> is necessarily kept small, and hence our efforts are directed at practical and accurate methodology for hypothesis testing for small <inline-formula><tex-math notation="LaTeX">$K.$</tex-math></inline-formula> Our generalized likelihood ratio statistic combined with exact cumulant matching gives very accurate rejection percentages. We also prove that the statistic on which the test is based is comprised of canonical coherencies arising from our complex-valued vector time series. Frequency specific tests are combined by using multiple hypothesis testing to give an overall test. Our methodology is demonstrated on ocean current data collected at different depths in the Labrador Sea. Overall, this paper extends results on propriety testing for complex-valued vectors to the complex-valued vector time series setting.