Chi-square and Poissonian Data: Biases Even in the High-Count Regime and How to Avoid them

TL;DR

This paper shows that common chi-square approximations in astronomical data fitting can cause biases even at high counts, recommending the use of Cash's C-statistic for unbiased parameter estimation.

Contribution

It demonstrates the biases of chi-square approximations in high-count regimes and advocates for using the C-statistic to avoid these biases in Poisson data fitting.

Findings

Chi-square approximations can cause significant biases in parameter estimates.

Cash's C-statistic provides more unbiased estimates at high counts.

Biases can impact satellite calibration and galaxy cluster measurements.

Abstract

We demonstrate that two approximations to the chi^2 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 counts 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 \sqrt{N_c}, where N_c is the total number of counts in the dataset, 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 chi^2 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 modelling real data. We illustrate the bias for two specific problems, measuring the count-rate from a lightcurve 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 mis-calibrated satellites and a ~5-10% shift in galaxy cluster scaling relations.