Laplace approximations for hypergeometric functions with Hermitian matrix argument

We develop highly accurate Laplace approximations for two hypergeometric functions of Hermitian matrix argument denoted by ₁F̃₁ and ₂F̃₁. These functions arise naturally in the multivariate noncentral distribution theory for complex multivariate normal models used in wireless communication, radar, sonar, and seismic detection. Each function arises as a factor in the moment generating function (MGF) for the noncentral distribution of the log-likelihood ratio statistic: ₁F̃₁ is a factor for complex MANOVA testing and ₂F̃₁ is a factor when testing block independence of complex normal signals. We use simulation to show the excellent accuracy achieved by the two approximations. We also compute ROC curves for these two tests by inverting the noncentral MGFs with saddlepoint approximations after replacing the true hypergeometric functions with their Laplace approximations.