Is the bump significant? An axion-search example

Many experiments in physics involve searching for a localized excess over background expectations in an observed spectrum. If the background is known and there is Gaussian noise, the amount of excess of successive observations can be quantified by the SQUARES (squares-of-run-residuals sum) statistic taking care of the look-elsewhere effect. The distribution of the SQUARES statistic under the background model is known analytically but the computation becomes too expensive for more than about a hundred observations. Here a principled high-precision extrapolation from a few dozen up to millions of data points is derived. It is most precise in the interesting regime when an excess is present. The method is verified for benchmark cases and successfully applied to real data from an axion search. C++ code that implements our method is available at https://github.com/fredRos/runs .