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 which
can be estimated by multitaper methods using, say,
K
tapers.
Standard asymptotic distributions for the test statistic are of no
use since they would require
K
→∞
,
but, as
K
increases so does
resolution bandwidth which causes spectral blurring. In many
analyses
K
is necessarily kept small, and hence our efforts are
directed at practical and accurate methodology for hypothesis
testing for small
K.
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 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 work extends results on propriety testing for complex-valued
vectors to the complex-valued vector time series setting.