Unbinned maximum-likelihood estimators for low-count data: Applications to faint X-ray spectra in the Taurus Molecular Cloud

Abstract

Traditional binned statistics such as $χ^2$ suffer from information loss and arbitrariness of the binning procedure. We point out that the underlying statistical quantity (the log likelihood $L$) does not require any binning beyond the one implied by instrumental readout channels, and we propose to use it for low-count data. The performance of $L$ in the model classification and point estimation problems is explored by Monte-Carlo simulations of Chandra and XMM X-ray spectra, and is compared to the performances of the binned Poisson statistic ($C$), Pearson's $χ^2$ and Neyman's $χ^2_N$, the Kolmogorov- Smirnov, and Kuiper' statistics. It is found that the unbinned log likelihood $L$ performs best with regard to the expected chi-square distance between true and estimated spectra, the chance of a successful identification among discrete candidate models, the area under the receiver-operator curve of reduced (two-model) binary classification problems, and generally also with regard to the mean square errors of individual spectrum parameters. The $χ^2$ ($χ^2_{\rm N}$) statistics should only be used if more than 10 (15) predicted counts per bin are available. From the practical point of view, the computational cost of evaluating $L$ is smaller than for any of the alternative methods if the forward model is specified in terms of a Poisson intensity and normalization is a free parameter. The maximum-$L$ method is applied to 14 observations from the Taurus Molecular Cloud, and the unbinned results are compared to binned XSPEC results, and found to generally agree, with exceptions explained by instability under re-binning and by background fine structures. The maximum-$L$ method has no lower limit on the available counts, and allows to treat weak sources which are beyond the means of binned methods.