χ²AND POISSONIAN DATA: BIASES EVEN IN THE HIGH-COUNT 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 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 √ 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 χ 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.

SNR 1E 0102.2-7219 as an X-ray calibration standard in the 0.5−1.0 keV bandpass and its application to the CCD instruments aboardChandra,Suzaku,SwiftandXMM-Newton

“…We fit the data in the energy range from 0.3−2.0 keV since that is the energy range in which E0102 dominates over the background. We adopted the C statistic as the fitting statistic to avoid the well-known bias with the χ 2 statistic with a low number of counts per bin (see Cash 1979;Nousek & Shue 1989) and the bias that persists even with a relatively large number of counts per bin (see Humphrey et al 2009). Given how bright E0102 is compared to the typical instrumental background, the low number of counts per bin bias should only affect the lowest and highest energies in the 0.3−2.0 keV bandpass.…”

Sterile neutrino dark matter bounds from galaxies of the Local Group

“…12.7.1 of the XSPEC spectral fitting package [63]. The data were mildly rebinned (to ensure >20 photons per bin), which aids in error bar computation, and we minimized the Cash-C goodness-offit statistic [64]. The model comprised a powerlaw and two thermal (APEC, [65]) plasma components to account for the cosmic and Galactic X-ray background, plus two broken power law models (not multiplied by the effective area) and three Gaussian lines to account for the instrumental background.…”

NuSTAROBSERVATIONS OF THE BULLET CLUSTER: CONSTRAINTS ON INVERSE COMPTON EMISSION

“…For continuum-driven fits on data binned to just above the Gaussian limit (25-30 counts bin −1 ), the χ 2 statistic is known to be biased, especially for fits using a large number of bins (Leccardi & Molendi 2007;Humphrey et al 2009). Briefly, the weights w on bins with negative fluctuations are overestimated while bins with positive fluctuations are underestimated, since w = 1/ √ N , so the χ 2 statistic drives the global best-fit model below the data.…”