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

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” (${\chi }_{\mathrm{C}\mathrm{N}\mathrm{P}}^2$). Through analytical derivations and toy model simulations, we show that ${\chi }_{\mathrm{C}\mathrm{N}\mathrm{P}}^2$ 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, ${\chi }_{\mathrm{C}\mathrm{N}\mathrm{P}}^2$ provides a good alternative given its natural connection to the covariance matrix formalism.