Locally Most Powerful Invariant Tests for Correlation and Sphericity of Gaussian Vectors

Details Locally Most Powerful Invariant Tests for Correlation and Sphericity of Gaussian Vectors Locally Most Powerful Invariant Tests for Correlation and Sphericity of Gaussian Vectors

2012-04-25

arxiv.org/abs/1204.5635v1

Computer Science

Information Theory

Scientific paper

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 much better performance than the typically used generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually requires 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 statistic distributions 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 {\it 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 provides better results than their GLRT counterparts.

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