Systematic errors in the maximum-likelihood regression of Poisson count data: introducing the overdispersed χ2 distribution

This paper presents a new method to estimate systematic errors in the maximum-likelihood regression of count data. The method is applicable in particular to X-ray spectra in situations where the Poisson log-likelihood, or the Cash goodness-of-fit statistic, indicate a poor fit that is attributable to overdispersion of the data. Overdispersion in Poisson data is treated as an intrinsic model variance that can be estimated from the best-fit model, using the maximum-likelihood C_min statistic. The paper also studies the effects of such systematic errors on the ΔC likelihood-ratio statistic, which can be used to test for the presence of a nested model component in the regression of Poisson count data. The paper introduces an overdispersed χ² distribution that results from the convolution of a χ² distribution that models the usual ΔC statistic, and a zero-mean Gaussian that models the overdispersion in the data. This is proposed as the distribution of choice for the ΔC statistic in the presence of systematic errors. The methods presented in this paper are applied to XMM-Newton data of the quasar 1ES 1553+113 that were used to detect absorption lines from an intervening warm-hot intergalactic medium (WHIM). This case study illustrates how systematic errors can be estimated from the data, and their effect on the detection of a nested component, such as an absorption line, with the ΔC statistic.