Mathematics – Statistics Theory
Scientific paper
2010-11-10
Annals of Statistics 2010, Vol. 38, No. 5, 2751-2780
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/10-AOS803 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/10-AOS803
This paper deals with recovering an unknown vector $\theta$ from the noisy data $Y=A\theta+\sigma\xi$, where $A$ is a known $(m\times n)$-matrix and $\xi$ is a white Gaussian noise. It is assumed that $n$ is large and $A$ may be severely ill-posed. Therefore, in order to estimate $\theta$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. For spectral regularization methods related to the so-called ordered smoothers [see Kneip Ann. Statist. 22 (1994) 835--866], we propose new penalties in the principle of empirical risk minimization. The heuristical idea behind these penalties is related to balancing excess risks. Based on this approach, we derive a sharp oracle inequality controlling the mean square risks of data-driven spectral regularization methods.
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