On virtues and vices of second-order measures of quality for binary classification

When deriving a detector, we are often led to consider design criteria such as second-order measures of quality. The aim of this paper is to provide a critical overview of these criteria. We first consider the case of deriving unconstrained detectors. We show that second-order criteria must satisfy a non-trivial condition to yield Bayes-optimal receivers, and thus to be considered as relevant criteria for detector design. Next, we address the case where constraints are imposed on the detection structure, leading us to consider some set C of admissible detectors. In these conditions we prove that even if it exists a monotonic function of the likelihood ratio in C, obtaining this statistic via the optimization of a second-order criterion, relevant or not, is not guaranteed. Finally, results are illustrated with simulation examples.