This paper analyzes a model in which an outcome equals a frontier function of inputs minus a nonnegative unobserved deviation. We allow the deviation’s distribution to depend on inputs, thereby allowing for endogeneity. If zero lies in the support of the deviation given inputs—an assumption we term assignment at the frontier—then the frontier is identified by the supremum of the outcome at those inputs, obviating the need for instrumental variables. We then estimate the frontier, allowing for random error whose distribution may also depend on inputs. Finally, we derive a lower bound on the mean deviation, using only variance and skewness, that is robust to a scarcity of data near the frontier. We apply our methods to estimate a firm-level frontier production function and inefficiency.