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Spatially adaptive splines for statistical linear inverse problems. (English) Zbl 1005.65053

Author’s 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 non-parametric settings, it is shown to attain optimal rates of convergence. Then its good performances are confirmed by means of a simulation study.

MSC:

65J10 Numerical solutions to equations with linear operators
62G07 Density estimation
65C60 Computational problems in statistics (MSC2010)
65J22 Numerical solution to inverse problems in abstract spaces
47A50 Equations and inequalities involving linear operators, with vector unknowns
65Y20 Complexity and performance of numerical algorithms
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