## The Annals of Statistics

### Risk hull method and regularization by projections of ill-posed inverse problems

#### Abstract

We study a standard method of regularization by projections of the linear inverse problem Y=Af+ε, where ε 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.

#### Article information

Source
Ann. Statist., Volume 34, Number 4 (2006), 1653-1677.

Dates
First available in Project Euclid: 3 November 2006

https://projecteuclid.org/euclid.aos/1162567628

Digital Object Identifier
doi:10.1214/009053606000000542

Mathematical Reviews number (MathSciNet)
MR2283712

Zentralblatt MATH identifier
1246.62082

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties

#### Citation

Cavalier, L.; Golubev, Yu. Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist. 34 (2006), no. 4, 1653--1677. doi:10.1214/009053606000000542. https://projecteuclid.org/euclid.aos/1162567628

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