Methods for $\chi^2$ parameter estimation using histograms

A common procedure in high energy physics data analysis is to derive correction factors to simulated events that make them better agree with data. This is often done using histograms created from data, and a Monte Carlo sample that is modified by some correction parameters. However, as the simulated events are modified, events will migrate between bins. This introduces discontinuities in the prediction: an infinitesimal perturbation of the correction factor can result in an event migrating across a bin boundary. As a result, a standard $\chi^2$ minimization will not work. This note discusses this problem and proposes a solution.