Application of maximal invariance to the ACE detection problem

Three simple adaptive detection statistics frequently appearing in the radar detection literature, which employ a sample covariance estimate for clutter suppression, are Kelly´s GLRT (generalized likelihood ratio test): the adaptive matched filter (AMF) and the adaptive cosine estimator (ACE), which is also a GLRT for the problem where the test-data power level is unconstrained relative to the training data. Bose and Steinhardt (1995, 1996) found a two-dimensional maximal invariant statistic for the adaptive detection problem for which Kelly´s statistic is a GLRT, a one-to-one function of the Kelly GLRT and the AMF. We extend this maximal-invariant framework to the adaptive detection problem for which ACE is a GLRT showing that ACE is a one-dimensional maximal invariant.