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In a recent paper, the second-order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing, which are associated with the two main types of quaternion properness. In this paper, the authors consider the problem of determining, from a finite number of independent vector observations, whether a quaternion Gaussian vector is proper or not. Specifically, they derive three Generalized Likelihood Ratio Tests (GLRTs) for testing the two main kinds of quaternion properness and show that the GLRTs reduce to the estimation of three previously proposed quaternion improperness measures. Interestingly, the three GLRT statistics (improperness measures) can be interpreted as an estimate of the entropy loss due to the quaternion improperness.
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