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PCA-kernel estimation

Author

Listed:
  • Biau Gérard
  • Mas André

    (Institut de Math´ematiques et de Modelisation de Montpellier, Equipe de Probabilites et Statistique, Montpellier Cedex 5, Frankreich)

Abstract

Many statistical estimation techniques for high-dimensional or functional data are based on a preliminary dimension reduction step, which consists in projecting the sample X1,...,Xn onto the first D eigenvectors of the Principal Component Analysis (PCA) associated with the empirical projector ^ ΠD. Classical nonparametric inference methods such as kernel density estimation or kernel regression analysis are then performed in the (usually small) D-dimensional space. However, the mathematical analysis of this data-driven dimension reduction scheme raises technical problems, due to the fact that the random variables of the projected sample (^ΠDX1,...,^ΠDXn) are no more independent. As a reference for further studies, we offer in this paper several results showing the asymptotic equivalencies between important kernel-related quantities based on the empirical projector and its theoretical counterpart. As an illustration, we provide an in-depth analysis of the nonparametric kernel regression case.

Suggested Citation

  • Biau Gérard & Mas André, 2012. "PCA-kernel estimation," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 19-46, March.
    • Handle: RePEc:bpj:strimo:v:29:y:2012:i:1:p:19-46:n:3
    DOI: 10.1524/strm.2012.1084
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    References listed on IDEAS

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    1. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
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