  In several interesting applications one is faced with the problem of
simultaneous binary hypothesis testing and parameter estimation. Although such
joint problems are not infrequent, there exist no systematic analysis in the
literature that treats them effectively. Existing approaches consider the
detection and the estimation subproblems separately, applying in each case the
corresponding optimum strategy. As it turns out the overall scheme is not
necessarily optimum since the criteria used for the two parts are usually
incompatible. In this article we propose a mathematical setup that considers
the two problems jointly. Specifically we propose a meaningful combination of
the Neyman-Pearson and the Bayesian criterion and we provide the optimum
solution for the joint problem. In the resulting optimum scheme the two parts
interact with each other, producing detection/estimation structures that are
completely novel. Notable side-product of our work is the proof that the well
known GLR test is finite-sample-size optimum under this combined sense.
