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Local linear–additive estimation for multiple nonparametric regressions

  • Lin, Lu
  • Song, Yunquan
  • Liu, Zhao
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    How to sufficiently use the structure information behind the data is still a challenging issue. In this paper, a local linear–additive estimation and its relevant version are proposed to automatically capture the additive information for general multiple nonparametric regressions. Our method connects two types of local estimators, the local linear (or the local constant) estimator and the local additive estimator. Thus the new estimators can achieve an adaptive fitting between the full model and the local (additive) model, and can adapt to the double additivity: local additivity and global additivity. On the other hand, like the local linear estimator, the new estimators can obtain the optimal convergence rate when the model has no additive structure. Moreover, the new estimators have closed representations and thus make the computation easy and accurate. The theoretical results and simulation studies show that the new approach has a low computational complexity and can significantly improve the estimation accuracy. Also a new theoretical framework is introduced as a foundation of locally and globally connected statistical inference. Based on this framework, the newly defined estimator can be regarded as a projection of the response variable onto full function space with respect to the locally and globally connected norms.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X13002042
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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 123 (2014)
    Issue (Month): C ()
    Pages: 252-269

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    Handle: RePEc:eee:jmvana:v:123:y:2014:i:c:p:252-269
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    1. Oliver Linton & E. Mammen & J. Nielsen, 1999. "The existence and asymptotic properties of a backfitting projection algorithm under weak conditions," LSE Research Online Documents on Economics 300, London School of Economics and Political Science, LSE Library.
    2. Juhyun Park & Burkhardt Seifert, 2010. "Local additive estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(2), pages 171-191.
    3. Jianqing Fan & Theo Gasser & Irène Gijbels & Michael Brockmann & Joachim Engel, 1997. "Local Polynomial Regression: Optimal Kernels and Asymptotic Minimax Efficiency," Annals of the Institute of Statistical Mathematics, Springer, vol. 49(1), pages 79-99, March.
    4. Lu Lin & Xia Cui & Lixing Zhu, 2009. "An Adaptive Two-stage Estimation Method for Additive Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 248-269.
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