The most common solution to the problem of comparing a pair of distribution functions (F, G) based on independent samples is to assume F and G belong to the Gaussian family with Var(F)=Var(G). Parametric assumptions such as these can be thought of as a mechanism for parametric smoothing. A Bayesian nonparametric procedure can be used as an option to full parametric smoothing. In this paper, we consider the situation where the Gaussian assumptions are believed to provide only an approximation to the true underlying distributions. We investigate the type of inferences which are available to practitioner in this simple, yet very important problem.