We propose a likelihood-based approach to Stochastic Non-Parametric Envelopment of Data (StoNED) estimator using a directional distance function with firm-specific directional vectors. Additionally, we show how to estimate firm-specific inefficiency estimates instead of focusing on their average only. Moreover, we propose models that are robust to misspecification in general and the use of unit-information-priors in this class of models. These priors control the amount of information to be exactly equal to one observation. In this context, we propose the use of Bayesian Bootstrapping to further mitigate possible misspecification. We also propose empirical tests for identification of the model. Monte Carlo experiments show the good performance of the new techniques and an empirical application to the technology of large U.S. banks shows the feasibility of the new techniques.