On equivalence between detectors obtained from second-order measures of performance

Second-order measures of performance are often used for deriving detectors. Many of these are related or equivalent. In this paper we propose unifying views of these criteria that depend on the approach used to design detectors. In the case of deriving unconstrained detectors, we show that second-order criteria can be classified into two categories depending on their ability to yield a one-to-one function of the likelihood ratio. Criteria that satisfy this fundamental property are logically called relevant criteria. If constraints are imposed on the detection structure, two relevant criteria may lead to non-equivalent detectors in the sense that the receiver operating characteristic is not the same. We then have to reconsider our strategy for partitioning the set of second-order measures of performance. For practical reasons, this problem is addressed in the case of linear detection structures. Within this context, we propose a necessary and sufficient condition under which distinct criteria provide equivalent detectors. Finally, results are illustrated by considering well-known second-order criteria.