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On the Estimation of a Support Curve of Indeterminate Sharpness

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

    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.

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    Bibliographic Info

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

    Volume (Year): 62 (1997)
    Issue (Month): 2 (August)
    Pages: 204-232

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

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    Related research

    Keywords: convergence rate curve estimation endpoint order statistic regular variation support;

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    Cited by:
    1. 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.
    2. 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.
    3. Daouia, Abdelaati & Noh, Hohsuk & Park, Byeong U., 2013. "Data envelope fitting with constrained polynomial splines," TSE Working Papers 13-449, Toulouse School of Economics (TSE).
    4. Klemelä, Jussi, 2004. "Complexity penalized support estimation," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 274-297, February.
    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, vol. 20(2), pages 311-333, August.
    6. 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.
    7. 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.

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