Estimation of additive frontier functions with shape constraints

Production frontier is an important concept in modern economics and has been widely used to measure production efficiency. Existing nonparametric frontier models often only allow one or low-dimensional input variables due to ‘curse-of-dimensionality’. In this paper we propose a flexible additive frontier model which quantifies the effects of multiple input variables on the maximum output. In addition, we consider the estimation of the nonparametric frontier functions with shape restrictions. Economic theory often imposes shape constraints on production frontier, such as, monotonicity and concavity. A two-step constrained polynomial spline method is proposed to give smooth estimates that automatically satisfy such shape constraints. The proposed method is not only easy to compute, but also more robust to outliers. In theory, we established uniform consistency of the proposed method. We illustrate the proposed method by both simulation studies and an application to the Norwegian farm data. The numerical studies suggest that the proposed method has superior performance by incorporating shape constraints.