A step forward on order-α robust nonparametric method: inclusion of weight restrictions, convexity and non-variable returns to scale

Partial frontiers have been recently developed in order to overcome several drawbacks of the traditional nonparametric techniques. These robust frontier (order-$$\alpha$$α and order-m) methods avoid the curse of dimensionality, are less sensitive to outliers and extreme data and may include direct environmental information in the model. Nonetheless, the disadvantages of these partial frontier-based methods according to the formulation proposed in the literature are that they do not allow weight restrictions or non-variable returns to scale technology. The procedure here proposed is an extension of the traditional order-$$\alpha$$α method, allowing the estimation of an empirical convex $$\alpha$$α-level, assuming also some additional constraints, such as the virtual weight restrictions and non-variable returns to scale. In the particular case of nonconvex attainable sets, unrestricted formulations and variable returns to scale assumption, the proposed procedure returns the same results as the standard order-$$\alpha$$α.