Use of Wijsman´s theorem for the ratio of maximal invariant densities in signal detection applications

We apply Wijsman´s theorem for the ratio of densities of the maximal invariant to signal detection applications. The method does not require the explicit use of a maximal invariant statistic or its density to derive the uniformly most powerful invariant (UMPI) test. We describe its use for common classes of detection problems and illustrate its use in examples. The analytic form of the representation provides new insight into the relationship between the UMPI and generalized likelihood ratio test.