Space-Time Separability in FMRI: Asymptotic Power Analysis and Cramér-Rao Lower Bounds

The space-time separability of covariance (equivalently spectrum) has been assumed and applied to most studies in functional magnetic resonance imaging (fMRI). However, until even very recently, almost no scientific justifications for it has been shown and its empirical validity has not been tested yet in fMRI. In a recent work, we developed a procedure for testing the space-time separability in the framework of the parametric cepstrum. In this correspondence, we provide two new contributions to the fMRI literature: 1) a derivation of the theoretical asymptotic power of the proposed separability test (as the numbers of voxels and time observations go to infinity) and 2) the derivation of an asymptotic Cramér-Rao lower bounds of signal and noise parameters. This analysis allows us to assess the impact of space-time non-separability on its detection power and/or the estimation accuracy of parametric cepstra, uncovering various important statistical properties of the proposed space-time separability test.