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Spatially Adaptive Splines for Statistical Linear Inverse Problems

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  • Cardot, Hervé

Abstract

This paper introduces a new nonparametric estimator based on penalized regression splines for linear operator equations when the data are noisy. A local roughness penalty that relies on local support properties of B-splines is introduced in order to deal with spatial heterogeneity of the function to be estimated. This estimator is shown to be consistent under weak conditions on the asymptotic behaviour of the singular values of the linear operator. Furthermore, in the usual nonparametric settings, it is shown to attain optimal rates of convergence. Then its good performances are confirmed by means of a simulation study.

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  • Cardot, Hervé, 2002. "Spatially Adaptive Splines for Statistical Linear Inverse Problems," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 100-119, April.
    • Handle: RePEc:eee:jmvana:v:81:y:2002:i:1:p:100-119
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    References listed on IDEAS

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    1. D. G. T. Denison & B. K. Mallick & A. F. M. Smith, 1998. "Automatic Bayesian curve fitting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 333-350.
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    Cited by:

    1. Cardot, Hervé & Sarda, Pacal, 2005. "Estimation in generalized linear models for functional data via penalized likelihood," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 24-41, January.
    2. Philip T. Reiss & R. Todd Ogden, 2009. "Smoothing parameter selection for a class of semiparametric linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 505-523, April.
    3. Dahmani, Abdelnasser & Ait Saidi, Ahmed & Bouhmila, Fatah & Aissani, Mouloud, 2009. "Consistency of the Tikhonov's regularization in an ill-posed problem with random data," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 722-727, March.

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