Combined Neyman–Pearson chi-square: An improved approximation to the Poisson-likelihood 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” ($X^2_{CNP}$). Through analytical derivations and toy model simulations, we show that 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, $X^2_{CNP}$ provides a good alternative given its natural connection to the covariance matrix formalism.