On the Invariance, Coincidence, and Statistical Equivalence of the GLRT, Rao Test, and Wald Test

Three common techniques to discriminate between alternatives in a binary hypothesis testing problem are: the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test. In this paper, we investigate some characteristics of the corresponding decision statistics and provide their expressions for some problems of particular interest in statistical signal processing. First of all, we focus on the invariance of the Rao and Wald tests with respect to transformations leaving the testing problem unaltered. Then, we introduce necessary and sufficient conditions in order for their decision statistics to coincide with twice the logarithm of the GLRT statistic. Finally, we present some detection problems, usually encountered in practical signal processing applications, where the decision variables of the three quoted tests are equivalent, namely related by strictly monotonic transformations.