Unbinned maximum-likelihood estimators for low-count data

Abstract

Traditional binned statistics such as χ 2 suffer from information loss and arbitrariness of the binning procedure, which is especially important at low count rates as encountered in the XMM-Newton Extended Survey of the Taurus Molecular Cloud (XEST). 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-Newton 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's 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 (twomodel) binary classification problems, and generally also with regard to the mean square errors of individual spectrum parameters. The χ 2 (χ 2 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 XEST observations, and confidence regions are discussed. 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. In particular, HO Tau is found by the unbinned method to be rather cool (kT ∼ 0.2 keV), which may be a sign of shock emission. 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.

χ2AND POISSONIAN DATA: BIASES EVEN IN THE HIGH-COUNT REGIME AND HOW TO AVOID THEM

“…In contrast, at least for some problems, fits using the C-statistic are found to be far less biased for low counts data (e.g. Nousek & Shue 1989;Churazov et al 1996;Arzner et al 2007), although not completely so (Leccardi & Molendi 2007).…”

Mass outflow of the X-ray emission line gas in NGC 4151

“…ARF and RMF files were generated for each OBSID, then the pairs of positive and negative spectra were combined for the first order spectra. The data were binned to a minimum of 10 counts per spectral bin such that the χ 2 statistic could be utilised (e.g., Arzner et al 2007). However, in order to determine whether our results were biased by our binning limit, we also fit the data binned to a minimum of 20 counts per spectral bin.…”

The ALHAMBRA survey: An empirical estimation of the cosmic variance for merger fraction studies based on close pairs

“…For example, Naylor & Jeffries (2006) used a MLE to fit colour-magnitude diagrams, Arzner et al (2007) to improve the determination of faint X-ray spectra, Makarov et al (2006) to improve distance estimates using red giant branch stars, and López-Sanjuan et al (2008, 2009a,b, 2010b) to estimate reliable merger fractions from morphological criteria. The MLEs are based on the estimation of the most probable values of a set of parameters, which define the probability distribution that describes an observational sample.…”