On a chi^2-function with previously estimated background

Abstract

There are intensive efforts searching for new phenomena in many present and future scientific experiments such as LHC at CERN, CLIC, ILC and many others. These new signals are usually rare and frequently contaminated by many different background events. Starting from the concept of profile likelihood we obtain what can be called a profile $χ^2$-function for counting experiments which has no background parameters to be fitted. Signal and background statistical fluctuations are automatically taking in account even when the content of some bins are zero. This paper analyzes the profile $χ^2$-function for fitting binned data in counting experiment when signal and background events obey Poisson statistics. The background events are estimated previously, either by Monte Carlo events, ``idle" run events or any other reasonable way. The here studied method applies only when the background and signal are completely independent events, i.e, they are non-coherent events. The profile $χ^2$-function has shown to have a fast convergence, with fewer events, to the ``true'' values for counting experiments as shown in MC toy tests. It works properly even when the bin contents are low and also when the signal to background ratio is small. Other interesting points are also presented and discussed. One of them is that the background parameter does not need to be estimated with very high precision even when there are few signal events during a fitting procedure. An application to Higgs boson discovery is discussed using previously published ATLAS/LHC experiment data.