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Extreme Values and Haar Series Estimates of Point Process Boundaries

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  • Stéphane Girard
  • Pierre Jacob

Abstract

ABSTRACT. We present a new method for estimating the edge of a two‐dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkably different possible limit distributions. We propose a method of reducing the negative bias, illustrated by a simulation.

Suggested Citation

  • Stéphane Girard & Pierre Jacob, 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, June.
    • Handle: RePEc:bla:scjsta:v:30:y:2003:i:2:p:369-384
    DOI: 10.1111/1467-9469.00336
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    Cited by:

    1. Girard, Stéphane, 2004. "On the asymptotic normality of the L1-error for Haar series estimates of Poisson point processes boundaries," Statistics & Probability Letters, Elsevier, vol. 66(1), pages 81-90, January.
    2. Girard, Stéphane & Guillou, Armelle & Stupfler, Gilles, 2013. "Frontier estimation with kernel regression on high order moments," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 172-189.
    3. 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.
    4. Beatriz Pateiro-López & Alberto Rodríguez-Casal, 2013. "Recovering the shape of a point cloud in the plane," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 19-45, March.
    5. 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.
    6. Girard, Séphane & Jacob, Pierre, 2009. "Frontier estimation with local polynomials and high power-transformed data," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1691-1705, September.
    7. 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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