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Cavalier, Laurent; Golubev, Yuri K.

Risk hull method and regularization by projections of ill-posed inverse problems. (English) Zbl 1246.62082

Ann. Stat. 34, No. 4, 1653-1677 (2006).
Summary: We study a standard method of regularization by projections of the linear inverse problem \(Y=Af+\epsilon \), where \(\epsilon\) is a white Gaussian noise, and \(A\) is a known compact operator with singular values converging to zero with polynomial decay. The unknown function \(f\) is recovered by a projection method using the singular value decomposition of \(A\). The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.

Cited in 1 Review
Cited in 40 Documents

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62M99 Inference from stochastic processes
65C60 Computational problems in statistics (MSC2010)

Keywords:

quadratic risk; oracle inequality
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Full Text: DOI arXiv Euclid

References:

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