A proposal for a different chi-square function for Poisson distributions

TL;DR

This paper introduces a new chi-square function based on an approximate Gaussian distribution derived from Poisson data, improving parameter estimation and goodness-of-fit testing for histograms, especially with small counts.

Contribution

It proposes a novel chi-square function for Poisson distributions that enhances curve fitting accuracy and convergence speed, particularly for small bin contents.

Findings

The new chi-square function performs well with small counts.

It converges to traditional methods with large data samples.

The method is simple to implement and computationally efficient.

Abstract

We obtain an approximate Gaussian distribution from a Poisson distribution after doing a change of variable. A new chi-square function is obtained which can be used for parameter estimations and goodness-of-fit testing when adjusting curves to histograms. Since the new distribution is approximately Gaussian we can use it even when the bin contents are small. The corresponding chi-square function can be used for curve fitting. This chi-square function is simple to implement and presents a fast convergence of the parameters to the correct value, especially for the parameters associated with the width of the fitted curve. We present a Monte Carlo comparative study of the fitting method introduced here and two other methods for three types of curves: Gaussian, Breit-Wigner and Moyal, when each bin content obeys a Poisson distribution. It is also shown that the new method and the other two converge to the same result when the number of events increases