Saddlepoint Approximations for Correlation Testing Among Multiple Gaussian Random Vectors

This letter considers the problem of threshold selection for a correlation test among multiple (>= 2) random vectors. The generalized likelihood ratio test (GLRT) for this problem uses a generalized Hadamard ratio to test for block diagonality in a composite covariance matrix. As the number of realizations used to estimate the composite covariance matrix grows large, the null distribution of the likelihood ratio statistic converges to a chi-squared distribution which can be used to prescribe thresholds needed to achieve a desired false alarm rate in high sample support situations. However, this asymptotic distribution can be slow to converge, making its use dubious in many practical scenarios. To address this problem, this letter uses saddlepoint approximations for the null distribution of the generalized Hadamard ratio. Simulations are provided to demonstrate the saddlepoint approximation's ability to achieve a desired false alarm probability, even in situations with low sample support.