Journal of Mathematics and Statistics

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

Fangyuan Nan

DOI : 10.3844/jmssp.2007.38.43

Journal of Mathematics and Statistics

Volume 3, Issue 2

Pages 38-43


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.


© 2007 Fangyuan Nan. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.