On the invariance property of 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 [F. Nan, R.D. Nowak, Generalized likelihood ratio detection for fMRI using complex data, IEEE Trans. Med. Imag. 18 (4) (1999) 320–329]. However, [F. Nan, R.D. Nowak, Generalized likelihood ratio detection for fMRI using complex data, IEEE Trans. Med. Imag. 18 (4) (1999) 320–329] is defective in the invariant study of the hypothesis testing problem: the invariant transformation group provided in Appendix II is not the largest and rigorous proof is not furnished either. This paper provides remedy to these deficiencies: the largest invariant transformation group is constructed rigorously and algebraically by matrix equation and method of undetermined coefficients. The group turns out to be larger than prescribed in [F. Nan, R.D. Nowak, Generalized likelihood ratio detection for fMRI using complex data, IEEE Trans. Med. Imag. 18 (4) (1999) 320–329]. Maximal invariants under both the original and the induced transformation groups are provided as well. The invariance property's utilities in exploring the dependence of the pdf on unknown parameters is strictly established. As a by-product of this paper, we point out that the geometrical method is defective in finding the largest invariant transformation group.