Abstract: Previous works have addressed the second-order statistical characterization of quaternion random vectors, introducing different properness definitions, and presenting the generalized likelihood ratio tests (GLRTs) for determining the kind of quaternion properness. This paper considers the more challenging problem of deriving the locally most powerful invariant tests (LMPITs), which can be obtained, even without an explicit expression for the maximal invariants, thanks to the Wijsman?s theorem. Specifically, we consider three binary hypothesis testing problems involving the two main kinds of quaternion properness, and show that the LMPIT statistics are given by the Frobenius norm of three previously defined sample coherence matrices. The proposed detectors exhibit interesting connections with the problem of testing for the properness of a complex vector, and with the problems of testing for the sphericity of a four-dimensional real (or two-dimensional complex proper) vector. Additionally, some numerical examples show that in general, the proposed LMPITs outperform their GLRT counterparts, and in some cases the performance gap is very noticeable.

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