# Combined Neyman-Pearson Chi-square: An Improved Approximation to the Poisson-likelihood Chi-square

## TL;DR

This paper introduces a new approximation called combined Neyman-Pearson chi-square that reduces bias in parameter estimation compared to traditional chi-square methods, offering a computationally efficient alternative to the Poisson-likelihood chi-square.

## Contribution

The paper proposes the combined Neyman-Pearson chi-square as an improved approximation to the Poisson-likelihood chi-square, with analytical and simulation validation.

## Key findings

- $ ext{CNP}$ chi-square reduces bias in parameter estimates.
- $ ext{CNP}$ provides a computationally efficient alternative.
- Significant bias reduction compared to Neyman's or Pearson's chi-square.

## Abstract

We describe an approximation to the widely-used Poisson-likelihood chi-square using a linear combination of Neyman's and Pearson's chi-squares, namely "combined Neyman-Pearson chi-square" ($\chi^2_{\mathrm{CNP}}$). Through analytical derivations and toy model simulations, we show that $\chi^2_\mathrm{CNP}$ leads to a significantly smaller bias on the best-fit model parameters compared to those using either Neyman's or Pearson's chi-square. When the computational cost of using the Poisson-likelihood chi-square is high, $\chi^2_\mathrm{CNP}$ provides a good alternative given its natural connection to the covariance matrix formalism.