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Local linear smoothing for regression surfaces on the simplex using Dirichlet kernels

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  • Christian Genest

    (McGill University)

  • Frédéric Ouimet

    (McGill University)

Abstract

This paper introduces a local linear smoother for regression surfaces on the simplex. The estimator solves a least-squares regression problem weighted by a locally adaptive Dirichlet kernel, ensuring good boundary properties. Asymptotic results for the bias, variance, mean squared error, and mean integrated squared error are derived, generalizing the univariate results of Chen (Ann Inst Stat Math, 54(2):312–323, 2002). A simulation study shows that the proposed local linear estimator with Dirichlet kernel outperforms its only direct competitor in the literature, the Nadaraya–Watson estimator with Dirichlet kernel due to Bouzebda et al. (AIMS Math 9(9):26195–26282, 2024).

Suggested Citation

  • Christian Genest & Frédéric Ouimet, 2025. "Local linear smoothing for regression surfaces on the simplex using Dirichlet kernels," Statistical Papers, Springer, vol. 66(4), pages 1-28, June.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:4:d:10.1007_s00362-025-01708-8
    DOI: 10.1007/s00362-025-01708-8
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    References listed on IDEAS

    as
    1. Song Chen, 2002. "Local Linear Smoothers Using Asymmetric Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(2), pages 312-323, June.
    2. Eduardo Fé, 2014. "Estimation and inference in regression discontinuity designs with asymmetric kernels," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(11), pages 2406-2417, November.
    3. Bertin, Karine & Genest, Christian & Klutchnikoff, Nicolas & Ouimet, Frédéric, 2023. "Minimax properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    4. Jianhong Shi & Weixing Song, 2016. "Asymptotic results in gamma kernel regression," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(12), pages 3489-3509, June.
    5. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    6. Axel Tenbusch, 1994. "Two-dimensional Bernstein polynomial density estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 41(1), pages 233-253, December.
    7. Bruce M. Brown & Song Xi Chen, 1999. "Beta‐Bernstein Smoothing for Regression Curves with Compact Support," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(1), pages 47-59, March.
    8. J. Aitchison & I. J. Lauder, 1985. "Kernel Density Estimation for Compositional Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 34(2), pages 129-137, June.
    9. Ouimet, Frédéric & Tolosana-Delgado, Raimon, 2022. "Asymptotic properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    10. Funke, Benedikt & Hirukawa, Masayuki, 2021. "Bias correction for local linear regression estimation using asymmetric kernels via the skewing method," Econometrics and Statistics, Elsevier, vol. 20(C), pages 109-130.
    Full references (including those not matched with items on IDEAS)

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