Application of the iterated weighted least-squares fit to counting experiments

Abstract Least-squares fits are popular in many data analysis applications, and so we review some theoretical results in regard to the optimality of this fit method. It is well-known that common variants of the least-squares fit applied to Poisson-distributed data produce biased estimates, but it is not well-known that the bias can be overcome by iterating an appropriately weighted least-squares fit. We prove that the iterated fit converges to the maximum-likelihood estimate. Using toy experiments, we show that the iterated weighted least-squares method converges faster than the equivalent maximum-likelihood method when the statistical model is a linear function of the parameters and it does not require problem-specific starting values. Both can be a practical advantage. The equivalence of both methods also holds for binomially distributed data. We further show that the unbinned maximum-likelihood method can be derived as a limiting case of the iterated least-squares fit when the bin width goes to zero, which demonstrates the deep connection between the two methods.