In this paper we consider discrimination between two possible sources based on observations of their output. The discrimination problem is modeled by means of a general binary hypothesis test, the main emphasis being on situations that cannot be modeled as signals in additive noise. The structure of the discriminator is such that the observations are passed through a memoryless nonlinearity and summed up to form a test statistic, which is then compared to a threshold. In this paper we consider only fixed sample size tests. Four different performance measures, which resemble the signal-to-noise ratios encountered in the signal in additive noise problems, are derived under different problem formulations. The optimal non-linearities for each of the performance measures are derived as solutions to various integral equations. For three of the four performance measures, we have successfully obtained robust nonlinearities for uncertainty in the marginal and the pint probaWlity density functions of the observations. Computer simulation results which demonstrate the advantage of using our non-linearities over the i.i.d. nonlinearity under the probability of error criterion are presented.