This paper analyzes a model in which an outcome variable equals a frontier function of inputs less a nonnegative unobserved deviation. If zero lies in the support of the deviation given inputs, then the frontier function is identified by the supremum outcome at those inputs. This obviates the need for instrumental variables. Implementation requires allowing for the distribution of deviations to depend on inputs, thus not ruling out endogeneity. Including random errors yields a stochastic frontier analysis model. We generalize this model to allow the joint distribution of deviations and errors to depend on inputs. We derive a lower bound for the mean deviation based only on variance and skewness, requiring no other distributional assumptions and remaining valid even when zero is not in the support of deviations or data are sparse near the frontier. We apply our results to a frontier production function, where deviations represent inefficiencies.

will be presented at the 1st International Conference on Applied Economics and Financial Issues, to be held in Thessaloniki on August 28–29, 2025