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Bandwidth choice for local polynomial estimation of smooth boundaries

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  • Hall, Peter
  • Park, Byeong U.

Abstract

Local polynomial methods hold considerable promise for boundary estimation, where they offer unmatched flexibility and adaptivity. Most rival techniques provide only a single order of approximation; local polynomial approaches allow any order desired. Their more conventional rivals, for example high-order kernel methods in the context of regression, do not have attractive versions in the case of boundary estimation. However, the adoption of local polynomial methods for boundary estimation is inhibited by lack of knowledge about their properties, in particular about the manner in which they are influenced by bandwidth; and by the absence of techniques for empirical bandwidth choice. In the present paper we detail the way in which bandwidth selection determines mean squared error of local polynomial boundary estimators, showing that it is substantially more complex than in regression settings. For example, asymptotic formulae for bias and variance contributions to mean squared error no longer decompose into monotone functions of bandwidth. Nevertheless, once these properties are understood, relatively simple empirical bandwidth selection methods can be developed. We suggest a new approach to both local and global bandwidth choice, and describe its properties.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:91:y:2004:i:2:p:240-261
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    References listed on IDEAS

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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. GIJBELS, Irène & MAMMEN, Enno & PARK, Byeong U. & SIMAR, Léopold, 1997. "On estimation of monotone and concave frontier functions," LIDAM Discussion Papers CORE 1997031, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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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. 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.
    3. 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.
    4. 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.
    5. Ebrahimi, Nader, 2008. "Simultaneous control of false positives and false negatives in multiple hypotheses testing," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 437-450, March.
    6. 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).
    7. Leonie Selk & Charles Tillier & Orlando Marigliano, 2022. "Multivariate boundary regression models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 400-426, March.
    8. Schick Anton & Wefelmeyer Wolfgang, 2009. "Non-standard behavior of density estimators for sums of squared observations," Statistics & Risk Modeling, De Gruyter, vol. 27(1), pages 55-73, November.

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