The robustness of the hyperbolic efficiency estimator

In this paper we examine the robustness properties of a specific type of orientation in the context of efficiency measurement using partial frontiers. This so called unconditional hyperbolic α-quantile estimator of efficiency has been recently studied by Wheelock and Wilson (2008) and can be seen as an extension of the in-put/output methodology of partial frontiers that was introduced by Aragon, Daouia and Thomas Agnan (2005). The influence function of this fully non-parametric and unconditional estimator is here derived for a complete multivariate setup (multiple inputs and outputs). Like for the input and output quantile estimators, the hyper-bolic α-quantile estimator is B-robust. The asymptotic variance of this estimator is recovered from the influence function. Some examples are given to assess the relevance of this type of estimator and to show the differences with the input and output α-quantile estimators of efficiency from both a robustness and a statistical efficiency point of view.