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Data envelope fitting with constrained polynomial splines

Listed author(s):
  • Abdelaati Daouia
  • Hohsuk Noh
  • Byeong U. Park

Estimation of support frontiers and boundaries often involves monotone and/or concave edge data smoothing. This estimation problem arises in various unrelated contexts, such as optimal cost and production assessments in econometrics and master curve prediction in the reliability programs of nuclear reactors. Very few constrained esti- mators of the support boundary of a bivariate distribution have been introduced in the literature. They are based on simple envelopment techniques which often suffer from lack of precision and smoothness. Combining the edge estimation idea of Hall, Park and Stern with the quadratic spline smoothing method of He and Shi, we develop a novel constrained fit of the boundary curve which benefits from the smoothness of spline approximation and the computational efficiency of linear programs. Using cubic splines is also feasible and more attractive under multiple shape constraints; computing the optimal spline smoother is then formulated into a second-order cone programming problem. Both constrained quadratic and cubic spline frontiers have a similar level of computational complexity to the unconstrained fits and inherit their asymptotic properties. The utility of this method is illustrated through applications to some real datasets and simulation evidence is also presented to show its superiority over the best known methods.

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File URL: http://hdl.handle.net/10.1111/rssb.12098
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Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society: Series B (Statistical Methodology).

Volume (Year): 78 (2016)
Issue (Month): 1 (January)
Pages: 3-30

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Handle: RePEc:bla:jorssb:v:78:y:2016:i:1:p:3-30
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  1. Hall, Peter & Park, Byeong U. & Stern, Steven E., 1998. "On Polynomial Estimators of Frontiers and Boundaries," Journal of Multivariate Analysis, Elsevier, vol. 66(1), pages 71-98, July.
  2. Hwang, J. H. & Park, B. U. & Ryu, W., 2002. "Limit theorems for boundary function estimators," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 353-360, October.
  3. Hardle, W. & Park, B. U. & Tsybakov, A. B., 1995. "Estimation of Non-sharp Support Boundaries," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 205-218, November.
  4. Jeong, Seok-Oh & Simar, Léopold, 2006. "Linearly interpolated FDH efficiency score for nonconvex frontiers," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2141-2161, November.
  5. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2009. "Frontier Estimation and Extreme Values Theory," IDEI Working Papers 611, Institut d'Économie Industrielle (IDEI), Toulouse.
  6. Kneip, Alois & Simar, Léopold & Wilson, Paul W., 2008. "Asymptotics And Consistent Bootstraps For Dea Estimators In Nonparametric Frontier Models," Econometric Theory, Cambridge University Press, vol. 24(06), pages 1663-1697, December.
  7. Hall, Peter & Park, Byeong U., 2004. "Bandwidth choice for local polynomial estimation of smooth boundaries," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 240-261, November.
  8. Hall, Peter & Nussbaum, Michael & Stern, Steven E., 1997. "On the Estimation of a Support Curve of Indeterminate Sharpness," Journal of Multivariate Analysis, Elsevier, vol. 62(2), pages 204-232, August.
  9. S.-O. Jeong & B. U. Park, 2006. "Large Sample Approximation of the Distribution for Convex-Hull Estimators of Boundaries," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 139-151.
  10. Tsybakov, A.B. & Korostelev, A.P. & Simar, L., 1992. "Efficient Estimation of Monotone Boundaries," Papers 9209, Catholique de Louvain - Institut de statistique.
  11. Stéphane Girard, 2003. "Extreme Values and Haar Series Estimates of Point Process Boundaries," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 369-384.
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