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Challenging the curse of dimensionality in multivariate local linear regression

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  • James Taylor
  • Jochen Einbeck

Abstract

Local polynomial fitting for univariate data has been widely studied and discussed, but up until now the multivariate equivalent has often been deemed impractical, due to the so-called curse of dimensionality. Here, rather than discounting it completely, we use density as a threshold to determine where over a data range reliable multivariate smoothing is possible, whilst accepting that in large areas it is not. The merits of a density threshold derived from the asymptotic influence function are shown using both real and simulated data sets. Further, the challenging issue of multivariate bandwidth selection, which is known to be affected detrimentally by sparse data which inevitably arise in higher dimensions, is considered. In an effort to alleviate this problem, two adaptations to generalized cross-validation are implemented, and a simulation study is presented to support the proposed method. It is also discussed how the density threshold and the adapted generalized cross-validation technique introduced herein work neatly together. Copyright Springer-Verlag 2013

Suggested Citation

  • James Taylor & Jochen Einbeck, 2013. "Challenging the curse of dimensionality in multivariate local linear regression," Computational Statistics, Springer, vol. 28(3), pages 955-976, June.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:3:p:955-976
    DOI: 10.1007/s00180-012-0342-0
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    References listed on IDEAS

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    1. Li, Qi & Racine, Jeff, 2003. "Nonparametric estimation of distributions with categorical and continuous data," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 266-292, August.
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    3. F. Ferraty & P. Hall & P. Vieu, 2010. "Most-predictive design points for functional data predictors," Biometrika, Biometrika Trust, vol. 97(4), pages 807-824.
    4. Sain, Stephan R., 2002. "Multivariate locally adaptive density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 165-186, April.
    5. Cumming, J.A. & Wooff, D.A., 2007. "Dimension reduction via principal variables," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 550-565, September.
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    Cited by:

    1. Zhou, Yunzhe & Shi, Chengchun & Li, Lexin & Yao, Qiwei, 2023. "Testing for the Markov property in time series via deep conditional generative learning," LSE Research Online Documents on Economics 119352, London School of Economics and Political Science, LSE Library.

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