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

TL;DR

This paper derives locally most powerful invariant tests for Gaussian vector covariance structures, showing they outperform traditional GLRT methods in detecting correlation and sphericity.

Contribution

It introduces explicit LMPITs for correlation and sphericity testing of Gaussian vectors using Wijsman's theorem, simplifying the derivation process and improving detection performance.

Findings

LMPIT for correlation is based on the Frobenius norm of the sample coherence matrix.

LMPIT for sphericity is based on the Frobenius norm of a normalized sample covariance matrix.

Numerical results show the proposed tests outperform GLRT in various scenarios.

Abstract

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.