  This paper studies the application of the generalized method of moments (GMM)
to multi-reference alignment (MRA): the problem of estimating a signal from its
circularly-translated and noisy copies. We begin by proving that the GMM
estimator maintains its asymptotic optimality for statistical models with group
symmetry, including MRA. Then, we conduct a comprehensive numerical study and
show that the GMM substantially outperforms the classical method of moments,
whose application to MRA has been studied thoroughly in the literature. We also
formulate the GMM to estimate a three-dimensional molecular structure using
cryo-electron microscopy and present numerical results on simulated data.
