On estimation of monotone and concave frontier functions

When analyzing the productivity of firms, one may want to compare how the firms transform a set of inputs I (typically labor, energy or capital) into an output y (typically a quantity of goods produced). The economic efficiency of a firm is then defined in terms of its ability to operate close to or on the production frontier, the boundary of the production set. The Frontier function gives the maximal level of output attainable by a firm for a given combination of its inputs. The efficiency of a firm may then be estimated via the distance between the attained production level and the optimal level given by the frontier function. From a statistical viewpoint, the frontier function may be viewed as the upper boundary of the support of the population of Alms density in the input and output space. It is often reasonable to assume that the production frontier is a concave monotone function. Then a famous estimator in the univariate input and output case is the data envelopment analysis (DEA) estimator, the lowest concave monotone increasing function covering all sample points. This estimator is biased downward, because it never exceeds the true production frontier. In this article we derive the asymptotic distribution of the DEA estimator, which enables us to assess the asymptotic bias and hence to propose an improved bias-corrected estimator. This bias-corrected estimator involves consistent estimation of the density function as well as of the second derivative of the production frontier. We also briefly discuss the construction of asymptotic confidence intervals. The finite-sample performance of the bias-corrected estimator is investigated via a simulation study, and the procedure is illustrated for a real data example.