Handling uncertainties in background shapes: the discrete profiling method

A common problem in high energy physics is the extraction of a signal over a potentially large background. In cases where the functional form of the background is not a priori known, then some extra uncertainty must be assigned to the signal parameters because of this lack of knowledge. A method for assigning an error from this cause is presented. It is based on treating the lack of knowledge of the functional form as a discrete nuisance parameter which is profiled in an equivalent way to continuous nuisance parameters. This effectively means an "envelope" is found which encompasses the lowest log-likelihood values for any given signal parameter. The bias and coverage of this method are shown to be good when applied to a realistic example.