Sufficient condition for an adaptive system to approximate the neyman-pearson detector

The application of adaptive systems to approximate the Neyman-Pearson detector is considered. The training error function is proved to be the key parameter that determines the possibility of approximating this detector. Based on the calculus of the approximated function for the selected error criterion, a sufficient condition is derived. Decision rules based on expressions of the optimum Bayes discriminant function, such as those approximated for the LMSE or the cross-entropy error criteria, have been analyzed. Previous works were based on the assumption that the system was trained to minimize the probability of error over the training set, so its performance was only optimal for the minimum probability of error threshold (system "operating point"). In this work, we prove that the decision rule based on the function approximated for an error function that fulfil the derived sufficient condition is optimum for all possible P_FA values. So, the concept of "operating point" will have no sense