χ2 and poissonian data: Biases even in the high-count regime and how to avoid them

Abstract

We demonstrate that two approximations to the χ2 statistic as popularly employed by observational astronomers for fitting Poisson-distributed data can give rise to intrinsically biased model parameter estimates, even in the high-count regime, unless care is taken over the parameterization of the problem. For a small number of problems, previous studies have shown that the fractional bias introduced by these approximations is often small when the counts are high. However, we show that for a broad class of problem, unless the number of data bins is far smaller than , where N c is the total number of counts in the data set, the bias will still likely be comparable to, or even exceed, the statistical error. Conversely, we find that fits using Cash's C-statistic give comparatively unbiased parameter estimates when the counts are high. Taking into account their well-known problems in the low-count regime, we conclude that these approximate χ2 methods should not routinely be used for fitting an arbitrary, parameterized model to Poisson-distributed data, irrespective of the number of counts per bin, and instead the C-statistic should be adopted. We discuss several practical aspects of using the C-statistic in modeling real data. We illustrate the bias for two specific problems - measuring the count rate from a light curve and obtaining the temperature of a thermal plasma from its X-ray spectrum measured with the Chandra X-ray observatory. In the context of X-ray astronomy, we argue the bias could give rise to systematically miscalibrated satellites and a ∼5-10% shift in galaxy cluster scaling relations. © 2009. The American Astronomical Society. All rights reserved..