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On the Estimation of a Support Curve of Indeterminate Sharpness


  • Hall, Peter
  • Nussbaum, Michael
  • Stern, Steven E.


We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:62:y:1997:i:2:p:204-232

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    References listed on IDEAS

    1. Tsybakov, A.B. & Korostelev, A.P. & Simar, L., 1992. "Efficient Estimation of Monotone Boundaries," Papers 9209, Catholique de Louvain - Institut de statistique.
    2. 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.
    3. Christensen, Laurits R & Greene, William H, 1976. "Economies of Scale in U.S. Electric Power Generation," Journal of Political Economy, University of Chicago Press, vol. 84(4), pages 655-676, August.
    4. Janssen, A. & Marohn, F., 1994. "On statistical information of extreme order statistics, local extreme value alternatives, and poisson point processes," Journal of Multivariate Analysis, Elsevier, vol. 48(1), pages 1-30, January.
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    Cited by:

    1. Abdelaati Daouia & Hohsuk Noh & Byeong U. Park, 2016. "Data envelope fitting with constrained polynomial splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 3-30, January.
    2. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2009. "Frontier Estimation and Extreme Values Theory," TSE Working Papers 10-165, Toulouse School of Economics (TSE).
    3. Goldenshluger, A. & Tsybakov, A., 2004. "Estimating the endpoint of a distribution in the presence of additive observation errors," Statistics & Probability Letters, Elsevier, vol. 68(1), pages 39-49, June.
    4. Terán, Pedro & López-Díaz, Miguel, 2014. "Strong consistency and rates of convergence for a random estimator of a fuzzy set," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 130-145.
    5. Abdelaati Daouia & Léopold Simar & Paul W. Wilson, 2017. "Measuring firm performance using nonparametric quantile-type distances," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 156-181, March.
    6. Daouia, Abdelaati & Girard, Stéphane & Guillou, Armelle, 2014. "A Γ-moment approach to monotonic boundary estimation," Journal of Econometrics, Elsevier, vol. 178(2), pages 727-740.
    7. Cheng, Ming-Yen & Hall, Peter, 2006. "Methods for tracking support boundaries with corners," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1870-1893, September.
    8. Girard, Stéphane & Jacob, Pierre, 2008. "Frontier estimation via kernel regression on high power-transformed data," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 403-420, March.
    9. repec:jss:jstsof:v:079:i09 is not listed on IDEAS
    10. Meister, Alexander, 2006. "Estimating the support of multivariate densities under measurement error," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1702-1717, September.
    11. 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;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 311-333, August.
    12. Klemelä, Jussi, 2004. "Complexity penalized support estimation," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 274-297, February.
    13. Daouia, Abdelaati & Laurent, Thibault & Noh, Hohsuk, 2017. "npbr: A Package for Nonparametric Boundary Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 79(i09).


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