Bayes-optimal detectors design using relevant second-order criteria

Statistical detection theories lead to the fundamental result that the optimum test consists in comparing any strictly monotone function of the likelihood ratio with a threshold value. In many applications, implementing such a test may be impossible. Therefore, we are often led to consider a simpler procedure for designing detectors. In particular, we can use alternative design criteria such as second-order measures of quality. In this paper, a necessary and sufficient condition is given for such criteria to guarantee the best solution in the sense of classical detection theories. This result is illustrated by discussing the relevance of well-known criteria.