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This paper considers the problem of determining whether a quaternion random vector is proper or not, which is an important problem because the structure of the optimal linear processing depends on the specific kind of properness. In particular, the authors focus on the Gaussian case and consider the two main kinds of quaternion properness, which yields three different binary hypothesis testing problems. The testing problems are solved by means of the Generalized Likelihood Ratio Tests (GLRTs) and the Locally Most Powerful Invariant Tests (LMPITs), which can be derived even without requiring an explicit expression for the maximal invariant statistics. Some simulation examples illustrate the performance of the proposed tests, which allows them to conclude that, for moderate sample sizes, it is advisable to use the LMPITs.
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