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

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