When is a maximal invariant hypothesis test better than the GLRT?

There has been considerable interest in applying maximal invariant (MI) hypothesis testing as an alternative to the generalized likelihood ratio test (GLRT). This interest has been motivated by several attractive theoretical properties of MI tests including: exact robustness to variation of nuisance parameters, finite-sample min-max optimality (in some cases), and distributional robustness, i.e. insensitivity to changes in the underlying probability distribution over a particular class. Furthermore, in some important cases the M test gives a reasonable test while the GLRT has worse performance than the trivial coin dip decision rule. However, in other cases, like the deep hide target detection problem, there are regimes (SNR, number of wireless users, coherence bandwidth) for which either of the MI and the GLRT can outperform the other. We discuss conditions under which the MI tests can be expected to outperform the GLRT in the context of a radar imaging and target detection application.