Testing for Quaternion Propriety

We consider the problem of testing whether a quaternion-valued Gaussian random vector is proper. The quaternion covariance matrix fully describes the second-order properties of a quaternion random vector only if the distribution is proper. The exact distribution of the likelihood ratio test under the hypothesis of propriety is derived for general sample size N and vector dimensionality p. As this is in terms of Meijer´s G-function, various approximations are considered, including Box-type and saddlepoint approximations. We find in particular that a new approach matching the first three cumulants is easy to implement and extremely accurate.