Transformation effects on invariant property of invariant hypothesis test and UMPI detector

The Uniformly Most Powerful (UMP) test, which is one with the highest probability of detection over all unknown parameters in a composite hypothesis test, does not exist in the most practical problems. It is common to derive the suboptimal Generalize Likelihood Ratio (GLR) detector, that is shown to perform close to optimal invariant tests in some problems. The Uniformly Most Powerful Invariant (UMPI) test is the one that is used for the performance evaluation of the GLR detector. However, the GLR or UMPI detectors are not simply derivable in many problems. In this paper, we suggest that the problem be transformed to a new space. We derive the conditions of the transformation, such that the detectors for the original problem and the transformed ones become equivalent. In addition, using this method, we discuss two practical problems (one in radar and other in communication) and derive the GLR detectors. We also find the UMPI detectors in these problems to verify the performance of the derived GLR detector.