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Estimation of Non-sharp Support Boundaries

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  • Hardle, W.
  • Park, B. U.
  • Tsybakov, A. B.

Abstract

Let X1, ..., Xn be independent identically distributed observations from an unknown probability density f(·), such that its support G = supp f is a subset of the unit square in 2. We consider the problem of estimating G from the sample X1, ..., Xn, under the assumption that the boundary of G is a function of smoothness [gamma] and that the values of density f decrease to 0 as the power [alpha] of the distance from the boundary. We show that a certain piecewise-polynomial estimator of G has optimal rate of convergence (namely, the rate n-[gamma]/(([alpha] + 1)[gamma] + 1)) within this class of densities.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 55 (1995)
Issue (Month): 2 (November)
Pages: 205-218

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Handle: RePEc:eee:jmvana:v:55:y:1995:i:2:p:205-218

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Cited by:
  1. Anderson, Gordon & Linton, Oliver & Whang, Yoon-Jae, 2012. "Nonparametric estimation and inference about the overlap of two distributions," Journal of Econometrics, Elsevier, Elsevier, vol. 171(1), pages 1-23.
  2. Daouia, Abdelaati & Noh, Hohsuk & Park, Byeong U., 2013. "Data envelope fitting with constrained polynomial splines," TSE Working Papers, Toulouse School of Economics (TSE) 13-449, Toulouse School of Economics (TSE).
  3. GIJBELS, Irène & MAMMEN, Enno & PARK, Byeong U. & SIMAR, Léopold, . "On estimation of monotone and concave frontier functions," CORE Discussion Papers RP -1392, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  4. Girard, Séphane & Jacob, Pierre, 2009. "Frontier estimation with local polynomials and high power-transformed data," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 100(8), pages 1691-1705, September.
  5. Abdelaati Daouia & Laurent Gardes & Stéphane Girard & Alexandre Lekina, 2011. "Kernel estimators of extreme level curves," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, Springer, vol. 20(2), pages 311-333, August.
  6. Girard, Stéphane & Jacob, Pierre, 2008. "Frontier estimation via kernel regression on high power-transformed data," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 99(3), pages 403-420, March.
  7. Jeong, Seok-Oh & Park, Byeong U., 2004. "Limit Distribution of Convex-Hull Estimators of Boundaries," Papers 2004,39, Humboldt-Universität Berlin, Center for Applied Statistics and Economics (CASE).
  8. Girard, Stéphane & Guillou, Armelle & Stupfler, Gilles, 2013. "Frontier estimation with kernel regression on high order moments," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 116(C), pages 172-189.
  9. U. Park, Byeong, 2001. "On estimating the slope of increasing boundaries," Statistics & Probability Letters, Elsevier, Elsevier, vol. 52(1), pages 69-72, March.
  10. Biau, Gérard & Cadre, Benoît & Pelletier, Bruno, 2008. "Exact rates in density support estimation," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 99(10), pages 2185-2207, November.
  11. Klemelä, Jussi, 2004. "Complexity penalized support estimation," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 88(2), pages 274-297, February.
  12. Goldenshluger, A. & Tsybakov, A., 2004. "Estimating the endpoint of a distribution in the presence of additive observation errors," Statistics & Probability Letters, Elsevier, Elsevier, vol. 68(1), pages 39-49, June.
  13. Hall, Peter & Park, Byeong U., 2004. "Bandwidth choice for local polynomial estimation of smooth boundaries," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 91(2), pages 240-261, November.
  14. Hwang, J. H. & Park, B. U. & Ryu, W., 2002. "Limit theorems for boundary function estimators," Statistics & Probability Letters, Elsevier, Elsevier, vol. 59(4), pages 353-360, October.
  15. Cheng, Ming-Yen & Hall, Peter, 2006. "Methods for tracking support boundaries with corners," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 97(8), pages 1870-1893, September.
  16. Daouia, Abdelaati & Girard, Stéphane & Guillou, Armelle, 2014. "A Γ-moment approach to monotonic boundary estimation," Journal of Econometrics, Elsevier, Elsevier, vol. 178(2), pages 727-740.

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