This paper considers a statistical model for a production frontier that is consistent with the traditional (nonstochastic) definition of a production function given in microeconomic theory. Limiting cases of the model are the familiar average production function and an envelope production function. Maximum-likelihood estimators for the parameters of the model are defined. The three related models are applied in the estimation of a production frontier for the Pastoral Zone of Eastern Australia with use of data from the Australian Grazing Industry Survey.