  We develop a new approach for estimating the risk of an arbitrary estimator
of the mean vector in the classical normal means problem. The key idea is to
generate two auxiliary data vectors, by adding carefully constructed normal
noise vectors to the original data. We then train the estimator of interest on
the first auxiliary vector and test it on the second. In order to stabilize the
risk estimate, we average this procedure over multiple draws of the synthetic
noise vector. A key aspect of this coupled bootstrap (CB) approach is that it
delivers an unbiased estimate of risk under no assumptions on the estimator of
the mean vector, albeit for a modified and slightly "harder" version of the
original problem, where the noise variance is elevated. We prove that, under
the assumptions required for the validity of Stein's unbiased risk estimator
(SURE), a limiting version of the CB estimator recovers SURE exactly. We then
analyze a bias-variance decomposition of the error of the CB estimator, which
elucidates the effects of the variance of the auxiliary noise and the number of
bootstrap samples on the accuracy of the estimator. Lastly, we demonstrate that
the CB estimator performs favorably in various simulated experiments.
