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Rates of Convergence for Spline Estimates of Additive Principal Components

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  • El Faouzi, Nour Eddin
  • Sarda, Pascal

Abstract

Additive principal components(APCs) generalize classicalprincipal component analysisto additive nonlinear transformations.Smallest APCsare additive functions of the vectorX=(X1, ..., Xp) minimizing the variance under orthogonality constraints and are characterized as eigenfunctions of an operator which is compact under a standard condition on the joint distribution of (X1, ..., Xp). As a by-product,smallest APCnearly satisfies the equation [summation operator]j [phi]j(Xj)=0 and then provides powerful tools for regression and data analysis diagnostics. The principal aim of this paper is the estimation of smallest APCs based on a sample from the distribution ofX. This is achieved using additive splines, which have been recently investigated in several functional estimation problems. The rates of convergence are then derived under mild conditions on the component functions. These rates are the same as the optimal rates for a nonparametric estimate of a univariate regression function.

Suggested Citation

  • El Faouzi, Nour Eddin & Sarda, Pascal, 1999. "Rates of Convergence for Spline Estimates of Additive Principal Components," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 120-137, January.
    • Handle: RePEc:eee:jmvana:v:68:y:1999:i:1:p:120-137
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    Cited by:

    1. Huang, Qiming & Zhu, Yu, 2016. "Model-free sure screening via maximum correlation," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 89-106.
    2. Salinelli, Ernesto, 2009. "Nonlinear principal components, II: Characterization of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 652-660, April.
    3. Jolliffe, Ian, 2022. "A 50-year personal journey through time with principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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