Testing for Quaternion Propriety
Paul Ginzberg, Andrew T. Walden · IEEE Transactions on Signal Processing · 2011
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 sizeNand vector dimensionalityp. 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.