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We consider the statistical inverse problem to recover f from noisy Y = Tf + \ \ where \ is Gaussian white noise and T compact operator between Hilbert spaces. Considering general reconstruction of the form \ f_\ = q_\ \ (T^*T\) T^*Y with an filter q_\, we investigate the choice of the regularization \ by minimizing an unbiased estimate of the predictive risk\ E\\ Tf - T\ f_\\²\. The corresponding \_ and its usage are well-known in the, but oracle inequalities and optimality results in this general are unknown. We prove a (generalized) oracle inequality, which relates direct risk \ E\\ f - \\_\²\ with the oracle prediction risk\\>₀\ E\\ Tf - T\ f\\²\. From oracle inequality we are then able to conclude that the investigated choice rule is of optimal order. Finally we also present numerical simulations, which support the order of the method and the quality of the parameter choice in finite situations.
Li et al. (Sun,) studied this question.