On Estimation of Monotone and Concave Frontier Functions

When analyzing the productivity of rms, one may want to compare how the rms transform a set of inputs x (typically labor, energy or capital) into an output y (typically a quantity of goods produced). The economic eeciency of a rm is then deened in terms of its ability of operating close to or on the production frontier which is the boundary of the production set. The frontier function gives the maximal level of output attainable by a rm for a given combination of its inputs. The eeciency of a rm may then be estimated via the distance between the attained production level and the optimal level given by the frontier function. From a statistical point of view, the frontier function may be viewed as the upper boundary of the support of the population of rms 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 which is the lowest concave monotone increasing function covering all sample points. This estimator is biased downwards since it never exceeds the true production frontier. In this paper 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 discuss brieey the construction of asymptotic conndence intervals. The nite sample performance of the bias corrected estimator is investigated via a simulation study and the procedure is illustrated for a real data example.