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Practical use of robust GCV and modified GCV for spline smoothing

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Abstract

Generalized cross-validation (GCV) is a popular parameter selection criterion for spline smoothing of noisy data, but it sometimes yields a severely undersmoothed estimate, especially if the sample size is small. Robust GCV (RGCV) and modified GCV are stable extensions of GCV, with the degree of stabilization depending on a parameter \(\gamma \in (0,1)\) for RGCV and on a parameter \(\rho >1\) for modified GCV. While there are favorable asymptotic results about the performance of RGCV and modified GCV, little is known for finite samples. In a large simulation study with cubic splines, we investigate the behavior of the optimal values of \(\gamma \) and \(\rho \), and identify simple practical rules to choose them that are close to optimal. With these rules, both RGCV and modified GCV perform significantly better than GCV. The performance is defined in terms of the Sobolev error, which is shown by example to be more consistent with a visual assessment of the fit than the prediction error (average squared error). The results are consistent with known asymptotic results.

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Authors and Affiliations

  1. Mathematics and Statistics, Murdoch University, South Street, Murdoch, WA, 6150, Australia

    Mark A. Lukas

  2. CSIRO Digital Productivity, GPO Box 664, Canberra, ACT, 2601, Australia

    Frank R. de Hoog & Robert S. Anderssen

Authors
  1. Mark A. Lukas
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  2. Frank R. de Hoog
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  3. Robert S. Anderssen
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Correspondence to Mark A. Lukas.

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Lukas, M.A., de Hoog, F.R. & Anderssen, R.S. Practical use of robust GCV and modified GCV for spline smoothing. Comput Stat 31, 269–289 (2016). https://doi.org/10.1007/s00180-015-0577-7

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  • DOI: https://doi.org/10.1007/s00180-015-0577-7

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