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Locally Most Powerful Invariant Tests for the Properness of Quaternion Gaussian Vectors

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Executive Summary

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, the authors 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.

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