The problem of estimating the center of symmetry of a symmetric signal in Gaussian white noise is considered. The underlying nuisance function f is not assumed to be differentiable, which makes a new point of view to the problem necessary.
We investigate the well-known sieve maximum likelihood estimators based on the cumulated periodogram, and study minimax rates over classes of irregular functions. It is shown that if the class appropriately controls the growth to infinity of the Fisher information over the sieve, semiparametric fast rates of convergence are obtained. We prove a lower bound result which implies that these semiparametric rates are really slower than the parametric ones, contrary to the regular case. Our results also suggest that there may be room to improve on the popular cumulated periodogram estimator.
"Semiparametric shift estimation based on the cumulated periodogram for non-regular functions." Electron. J. Statist. 5 102 - 126, 2011. https://doi.org/10.1214/11-EJS599