Application of the Iterated Weighted Least-Squares Fit to counting experiments

TL;DR

This paper demonstrates that iterated weighted least-squares fitting can correct bias in counting experiment data, aligning results with maximum-likelihood estimates and offering practical advantages over traditional methods.

Contribution

It introduces an iterated weighted least-squares approach that converges faster and is more practical than maximum-likelihood fitting for counting data analysis.

Findings

Iterated weighted least-squares produces unbiased estimates for Poisson data.

The method converges faster than maximum-likelihood for linear models.

Unbinned maximum-likelihood is a limit case of the iterated least-squares fit.

Abstract

Least-squares fits are an important tool in many data analysis applications. In this paper, we review theoretical results, which are relevant for their application to data from counting experiments. Using a simple example, we illustrate the well known fact that commonly used variants of the least-squares fit applied to Poisson-distributed data produce biased estimates. The bias can be overcome with an iterated weighted least-squares method, which produces results identical to the maximum-likelihood method. For linear models, the iterated weighted least-squares method converges faster than the equivalent maximum-likelihood method, and does not require problem-specific starting values, which may 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 a deep connection between the two methods.