In this paper, we study the existence of locally most powerful invariant tests (LMPIT) for the problem of testing the covariance structure of a set of Gaussian random vectors. The LMPIT is the optimal test for the case of close hypotheses, among those satisfying the invariances of the problem, and in practical scenarios can provide better performance than the typically used generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually requires one to find the maximal invariant statistic for the detection problem and then derive its distribution under both hypotheses, which in general is a rather involved procedure. As an alternative, Wijsman's theorem provides the ratio of the maximal invariant densities without even finding an explicit expression for the maximal invariant. We first consider the problem of testing whether a set of N-dimensional Gaussian random vectors are uncorrelated or not, and show that the LMPIT is given by the Frobenius norm of the sample coherence matrix. Second, we study the case in which the vectors under the null hypothesis are uncorrelated and identically distributed, that is, the sphericity test for Gaussian vectors, for which we show that the LMPIT is given by the Frobenius norm of a normalized version of the sample covariance matrix. Finally, some numerical examples illustrate the performance of the proposed tests, which provide better results than their GLRT counterparts.
@cite_8: The majority of test criteria used in multivariate analysis are functions of the roots of certain determinantal equations constructed from sample moments of order not greater than two. In general there exists no best test. Pillai (1966), Pillai & Jayachandran (1967, 1968) and Pillai & Dotson (1969) have compared the powers of a few among many possible competing test criteria by evaluating their powers numerically in small dimensional cases. The numerical approach, besides being costly and laborious, has a number of other obvious limitations. That locally best tests can be derived mathematically from known results seems not to have been noted.
In this paper, we focus on the case of close hypotheses and study the existence of LMPITs for the covariance structure of Gaussian data. We extend @cite_8 to the general case of vector-valued observations that may or may not have the same covariance matrix under the null hypothesis. Instrumental in deriving these results is the application of Wijsman's theorem, which requires identifying the invariances of each problem and integrating the distribution of the transformed observations under a measure on the corresponding group of transformations. Although we focus on the case of complex Gaussian vectors, which is motivated by its applications in radar and spectrum sensing problems, the presented results can be easily proved for real vectors, which is left as an exercise for the interested reader. Specifically, the main contributions of this work are the following: