Finite Sample Size Optimality of GLR Tests

In binary hypothesis testing, when the hypotheses are composite or the corresponding data pdfs contain unknown parameters, one can use the well known generalized likelihood ratio test (GLRT) to reach a decision. This test has the very desirable characteristic of performing simultaneous detection and estimation in the case of parameterized pdfs or combined detection and isolation in the case of composite hypotheses. Although GLRT is known for many years and has been the decision tool in numerous applications, only asymptotic optimality results are currently available to support it. In this work a novel, finite sample size, detection/estimation formulation for the problem of hypothesis testing with unknown parameters and a corresponding detection/isolation setup for the case of composite hypotheses, is introduced. The resulting optimum scheme has a GLRT-like form which is closely related to the criterion one adopts for the parameter estimation or isolation part. When this criterion is selected in a very specific way we recover the well known GLRT of the literature while interesting novel tests are obtained with alternative criteria. The mathematical derivations are surprisingly simple considering they solve a problem that has been open for more than half a century.