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Kernel Estimation: the Equivalent Spline Smoothing Method

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  • Härdle, Wolfgang Karl
  • Nussbaum, Michael

Abstract

Among nonparametric smoothers, there is a well-known correspondence between kernel and Fourier series methods, pivoted by the Fourier transform of the kernel. This suggests a similar relationship between kernel and spline estimators. A known special case is the result of Silverman (1984) on the effective kernel for the classical Reinsch-Schoenberg smoothing spline in the nonparametric regression model. We present an extension by showing that a large class of kernel estimators have a spline equivalent, in the sense of identical asymptotic local behaviour of the weighting coefficients. This general class of spline smoothers includes also the minimax linear estimator over Sobolev ellipsoids. The analysis is carried out for piecewise linear splines and equidistant design.

Suggested Citation

  • Härdle, Wolfgang Karl & Nussbaum, Michael, 2020. "Kernel Estimation: the Equivalent Spline Smoothing Method," IRTG 1792 Discussion Papers 2020-010, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    • Handle: RePEc:zbw:irtgdp:2020010
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    Keywords

    Kernel estimator; spline smoothing; filtering coefficients; differential operator; Green's function approximation; asymptotic minimax spline;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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