Detection of a small shift in a broad distribution

Statistical methods for the extraction of a small shift in broad data distributions are examined by means of Monte Carlo simulations. This work was originally motivated by the CERN neutrino beam to Gran Sasso (CNGS) experiment for which the OPERA detector collaboration reported a time shift in a broad distribution with an accuracy of $\pm 7.8\,$ns, while the fluctuation of the average time turns with $\pm 23.8\,$ns out to be much larger. Although the physical result of a big shift has been withdrawn, statistical methods that make an identification in a broad distribution with such a small error possible remain of interest.