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A large amount of literature has been developed on how to specify and to estimate production frontiers or cost functions. Two different approaches have been mainly developed: the deterministic frontier model which relies on the assumption that all the observations are on a unique side of the frontier, and the stochastic frontier models where observational errors or random noise allows some observations to be outside of the frontier. In a deterministic frontier framework, nonparametric methods are based on envelopment techniques known as FDH (Free Disposal Hull) and DEA (Data Envelopment Analysis). Today, statistical inference based on DEA/FDH type of estimators is available but, by construction, they are very sensitive to extreme values and to outliers. In this paper, we build an original nonparametric estimator of the “efficient frontier” which is more robust to extreme values, noise or outliers than the standard DEA/FDH nonparametric estimators. It is based on a concept of expected minimum input function (or expected maximal output function). We show how these functions are related to the efficient frontier itself. The resulting estimator is also related to the FDH estimator but our estimator will not envelop all the data. The asymptotic theory is also provided. Our approach includes the multiple inputs and multiple outputs cases. As an illustration, the methodology is applied to estimate the expected minimum cost function for french post offices.