On Testing for Impropriety of Complex-Valued

We consider the problem of testing whether a com- plex-valued random vector is proper, i.e., is uncorrelated with its complex conjugate. We formulate the testing problem in terms of real-valued Gaussian random vectors, so we can make use of some useful existing results which enable us to study the null distribu- tions of two test statistics. The tests depend only on the sample-size and the dimensionality of the vector . The basic behaviors of the distributions of the test statistics are derived and critical values (thresholds) are calculated and presented for certain values. For one of these tests we derive a distributional approximation for a transform of the statistic, potentially very useful in practice for rapid and simple testing. We also study the power (detection prob- ability) of the tests. Our results mean that testing for propriety can be a practical and undaunting procedure.

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