On the Probability Density Function of the Test Statistic for One Nonlinear GLR Detector Arising from fMRI

Recently an important and interesting nonlinear generalized likelihood ratio (GLR) detector emerged in functional magnetic resonance imaging (fMRI) data processing. However, the study of that detector is incomplete: the probability density function (pdf) of the test statistic was draw from numerical simulations without much theoretical support and is therefore, not firmly grounded. This correspondence presents more accurate (asymptotic) closed form of the pdf by resorting to a non-central Wishart matrix and by asymptotic expansion of some integrals. It is then confirmed theoretically that the detector does possess constant false alarm rate (CFAR) property under some practical regimes of signal to noise ratio (SNR) for finite samples and the correct threshold selection method is given, which is very important for real fMRI data processing.