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On Estimation of Monotone and Concave Frontier Functions

Gijbels, I. ; Park, Byeong U. ; Mammen, Enno ; Simar, Leopold

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Official URL: urn:nbn:de:kobv:11-10056435
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Abstract

When analyzing the productivity of firms, one may want to compare how the firms transform a set of inputs x (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 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 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 point of view, the frontier function may be viewed as the upper boundary of the support of the population of firms 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 briefly 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.

Item Type: Working paper
Place of Publication: Heidelberg
Date Deposited: 07 Jun 2016 07:31
Date: 1 August 1997
Number of Pages: 25
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 510 Mathematics
Controlled Keywords: Asymptotische Verteilung
Uncontrolled Keywords: Confidence interval, Asymptotic distribution, bias correction, data envelopment analysis density support, frontier function
Schriftenreihe ID: Beiträge zur Statistik > Beiträge
Additional Information: Erschienen auch in: Discussion Papers, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes 1998,9
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