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

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.

### MSC:

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

### Keywords:

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