Artur Prędki

(Polish) PDF


In the paper estimation methods of the production function which are used in the special frontier model are described. The frontier model is of a semiparametric nature, as the production frontier is not given analytically. However, assumptions related to the model errors have parametric character. In order to estimate the frontier, the so-called StoNED (Stochastic Non-smooth Envelopment of Data) multi-stage procedure is used. The first step forms the CNLS method (Convex Nonparametric Least Squares), which is a nonparametric counterpart of OLS. Next stages run according to a scheme, which has been used on the ground of the stochastic frontier models for many years. As a result not only the estimator of the production function is received, but also the estimator of the inefficiency term for a given production unit. In the paper some critical remarks are presented. They are related to faults and limitations of the research methodology. Next, the procedure is illustrated with an empirical study. Finally, potential directions for further research in the area are presented.


semiparametric frontier model, production function, CNLS method, StoNED method


[1] Afriat S.N., (1967), The construction of a utility function from expenditure data, “International Economic Review” Vol. 8, 67-77.

[2] Aigner D., Lovell C.A.K., Schmidt P., (1977), Formulation and estimation of stochastic frontier models, “Journal of Econometrics” Vol. 6, 21-37.

[3] Banker R.D., (1993), Maximum Likelihood, Consistency and Data Envelopment Analysis: A Statistical Foundation, “Management Science” Vol. 39 (No. 10), 1265-1273.

[4] Fan Y., Li Q., Weersink A., (1996), Semiparametric estimation of stochastic production frontier models, “Journal of Business and Economic Statistics” Vol. 14 (No 4), 460-468.

[5] Fraser D.A.S., Massam H., (1989), A mixed primal-dual bases algorithm for regression under inequality constraints: Application to concave regression, “Scandinavian Journal of Statistics” Vol. 16, 65-74.

[6] Greene W.H., (2008), The econometric approach to efficiency analysis, rozdział 2 w: Fried H., Lovell K., Schmidt S. (eds) “The measurement of productive efficiency and productivity growth”, Oxford University Press, New York.

[7] Groeneboom P., Jongbloed G., Wellner J.A., (2001), Estimation of a convex function: characterizations and asymptotic theory, “Annals of Statistics” Vol. 29, 1653-1698.

[8] Hanson D.L., Pledger G., (1976), Consistency in concave regression, “Annals of Statistics” Vol. 4 (No 6), 1038-1050.

[9] Hildreth C., (1954), Point estimates of ordinates of concave functions, “Journal of the American Statistical Association” Vol. 49, 598-619.

[10] Horrace W.C., Schmidt P., (1996), Confidence statements for efficiency estimates from stochastic frontier models, “Journal of Productivity Analysis” Vol. 7, 257-282.

[11] Jondrow J., Lovell C.A.K., Materov I.S., Schmidt P., (1982), On estimation of technical inefficiency in the stochastic frontier production function model, “Journal of Econometrics” Vol. 19, 233-238.

[12] Kumbhakar S.C., Lovell C.A.K., (2000), Stochastic frontier analysis, Cambridge University Press, Cambridge.

[13] Kuosmanen T., (2006), Stochastic nonparametric envelopment of data: combining virtues of SFA and DEA in a unified framework, MTT Discussion Paper 3/2006 (dostepny na stronie:

[14] Kuosmanen T., (2008), Representation theorem for convex nonparametric least squares, “Journal of Econometrics” Vol. 11, 308-325.

[15] Kuosmanen T., Johnson A., (2010), Data envelopment analysis as nonparametric least squares regression, “Operations Research” Vol. 58 (No 1), 149-160.

[16] Kuosmanen T., Kortelainen M., (2012), Stochastic non-smooth envelopment of data: semiparametric frontier estimation subject to shape constraints, “Journal of Productivity Analysis” Vol. 38, 11-28.

[17] Kuosmanen T., Kuosmanen N., (2009), Role of benchmark technology in sustainable value analysis: an application to Finnish dairy farms, “Agricultural and Food Science” Vol. 18, 302-316.

[18] Osiewalski J., Wróbel-Rotter R., (2002), Bayesowski model efektów losowych w analizie efektywnosci kosztowej (na przykładzie elektrowni i elektrociepłowni polskich), „Przeglad Statystyczny” Vol. 50 (No 2), 47-68.

[19] Predki A., (2003), Analiza efektywnosci za pomoca metody DEA: podstawy formalne i ilustracja ekonomiczna, „Przeglad Statystyczny” Vol. 50 (No 1), 87-100.

[20] Predki A., (2011), Spojrzenie na metody estymacji w modelach regresyjnych przez pryzmat programowania matematycznego, wysłane do druku w Wyd. UE we Wrocławiu.

[21] Richmond J., (1974), Estimating the efficiency of production, “International Economic Review” Vol. 15 (No 2), 515-521.

[22] Serfling R.J., (1991), Twierdzenia graniczne statystyki matematycznej, PWN Warszawa.

[23] Simar L., Wilson P.W., (2008), Statistical Inference in Nonparametric Frontier Models: Recent Developments and Perspectives, rozdział 4 w: Fried H., Lovell K., Schmidt S. (eds) “The measurement of productive efficiency and productivity growth”, Oxford University Press, New York.

[24] Simar L.,Wilson P.W., (2010), Estimation and inference in crosssectional stochastic frontier models,“Econometric Reviews” Vol. 29 (No 1), 62-98.

[25] Varian H., (1982), The nonparametric approach to demand analysis, “Econometrica” Vol. 50, 945-973.

[26] Wu C.F., (1982), Some algorithms for concave and isotonic regression, “TIMS Studies in Management Science” Vol. 19, 105-116.

[27] Yatchew A.J., Bos L., (1997), Nonparametric regression and testing in economic models, “Journal of Quantitative Economics” Vol. 13, 81-131.

Back to top
Copyright © 2019 Statistics Poland