χ^{2} and Poissonian Data: Biases Even in the HighCount Regime and How to Avoid Them
Abstract
We demonstrate that two approximations to the χ^{2} statistic as popularly employed by observational astronomers for fitting Poissondistributed data can give rise to intrinsically biased model parameter estimates, even in the highcount 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 √{N_c}, 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 Cstatistic give comparatively unbiased parameter estimates when the counts are high. Taking into account their wellknown problems in the lowcount regime, we conclude that these approximate χ^{2} methods should not routinely be used for fitting an arbitrary, parameterized model to Poissondistributed data, irrespective of the number of counts per bin, and instead the Cstatistic should be adopted. We discuss several practical aspects of using the Cstatistic 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 Xray spectrum measured with the Chandra Xray observatory. In the context of Xray astronomy, we argue the bias could give rise to systematically miscalibrated satellites and a ~510% shift in galaxy cluster scaling relations.
 Publication:

The Astrophysical Journal
 Pub Date:
 March 2009
 DOI:
 10.1088/0004637X/693/1/822
 arXiv:
 arXiv:0811.2796
 Bibcode:
 2009ApJ...693..822H
 Keywords:

 methods: data analysis;
 methods: statistical;
 Xrays: galaxies: clusters;
 Xrays: general;
 Astrophysics
 EPrint:
 9 pages, 2 figures. Accepted for publication in the Astrophysical Journal