  We consider current and alternative approaches to setting limits on new
physics signals having backgrounds from misidentified objects; for example jets
misidentified as leptons, b-jets or photons. Many ATLAS and CMS analyses have
used a heuristic matrix method for estimating the background contribution from
such sources. We demonstrate that the matrix method suffers from statistical
shortcomings that can adversely affect its ability to set robust limits. A
rigorous alternative method is discussed, and is seen to produce fake rate
estimates and limits with better qualities, but is found to be too costly to
use. Having investigated the nature of the approximations used to derive the
matrix method, we propose a third strategy that is seen to marry the speed of
the matrix method to the performance and physicality of the more rigorous
approach.
