ARTICLE
ABSTRACT
In the paper some general statistical model is presented and one of its particular version is exploited to formal estimate of the production set within the DEA and FDH methods. Properties of the FDH and DEA estimators are presented and their realizations for a finite sample are illustrated. Elements of the minimax approach are introduced and the rate of convergence is exploited to express the definition of asymptotic optimality of the estimators.
KEYWORDS
nonparametric statistical model, minimax approach, rate of convergence, production set, set estimation, FDH method, DEA method
REFERENCES
[1] Banker R.D., [1993], Maximum Likelihood, Consistency and Data Envelopment Analysis: A Statistical Foundation, Management Science, Vol. 39 (No. 10), s. 1265-1273.
[2] Banker R.D., Charnes A., Cooper W.W., [1984], Some models for estimating technical and scale inefficiencies in DEA, Management Science, Vol. 30 (No. 9), s. 1078-1091.
[3] Chevalier J., [1976], Estimation du support et du contenu du support d’une loi de probabilité, Annals Inst. H. Poincaré Sec. B Vol. 12, s. 339-364.
[4] Cuevas A., Rodriguez-Casal A., [2003], Set estimation: an overview and some recent developments, in „Recent Advances and Trends in Nonparametric Statistics” (red. Akritas, Politis), s. 251-264 North-Holland, Amsterdam.
[5] Deprins D., Simar L., Tulkens H., [1984], Measuring labor efficiency in post offices, in „The Performance of Public Enterprises: Concepts and Measurements” (red. Marchand, Pestieau, Tulkens), s. 243-267, Amsterdam, North-Holland.
[6] Devroye L., Wise L.G., [1980], Detection of abnormal behavior via nonparametric estimation of the support, SIAM J.Appl. Math. Vol. 38, s. 480-488.
[7] Dümbgen L., Walther G., [1996], Rates of convergence for random approximations of convex sets, Adv. Appl. Prob. Vol. 28, s. 384-393.
[8] Ibragimov I.A., Khasminskii R.Z., [1981], Statistical Estimation: Asymptotic Theory, Springer-Verlag, New York.
[9] Kleiner W., [1986], Analiza matematyczna, tom I, PWN Warszawa.
[10] Korostelev A.P., Simar L., Tsybakov A.B., [1995], Efficient estimation of monotone boundaries, The Annals of Statistics Vol. 23, s. 476-489.
[11] Korostelev A.P., Simar L., Tsybakov A.B., [1995], On estimation of monotone and convex boundaries, Publications de l’Institut de Statistique des Universités de Paris XXXIX Vol. 1, s. 3-18.
[12] Korostelev A.P., Tsybakov A.B., [1993], Minimax Theory of Image Reconstruction, Springer-Verlag, New York.
[13] Ombach J., [1993], Wstęp do rachunku prawdopodobieństwa, Wyd. UJ, Kraków.
[14] Pajor A., Prędki A., [2009], Nieparametryczna estymacja miernika efektywności technicznej w ramach metody DEA, Przegląd Statystyczny Vol. 56 (No. 3-4), s. 5-25.
[15] Prędki A., [2006], Definiowanie globalnego i lokalnego efektu skali w ramach badania efektywności metodą DEA, Przegląd Statystyczny Vol. 53 (No. 3), s. 57-72.
[ 16] Prędki A., Propozycja opisu niepewności w ramach metod DEA i FDH, wysłane do druku w Wyd. Uniwersytetu Wrocławskiego.
[17] Rényi A., Sulanke R., [1963], Über die konvexe Hülle von n zufällig gewählten Punkten, Z. Wahrschei nlichkeitstheorie Verw. Gebiete Vol. 2, s. 75-84.
[18] Rényi A., Sulanke R., [1964], Über die konvexe Hülle von n zufällig gewählten Punktem II, Z. Wahrsc heinlichkeitstheorie Verw. Gebiete Vol. 3, s. 138-147.
[19] Schneider R., [1988], Random approximation of convex sets, J. Microscopy Vol. 151, s. 211-227.
[20] Serfling R.J., [1991], Twierdzenia graniczne statystyki matematycznej, PWN Warszawa.
[21] Simar L., Wilson W., [2008], Statistical Inference in Nonparametric Frontier Models: Recent Developments and Perspectives, in “The Measurement of Productive Efficiency and Productivity Growth” (red. Fried, Lovell, Schmidt), Oxford University Press.
