Unbinned maximumlikelihood estimators for lowcount data. Applications to faint Xray spectra in the Taurus molecular cloud
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 XMMNewton 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 lowcount data. The performance of L in the model classification and point estimation problems is explored by MonteCarlo simulations of Chandra and XMMNewton Xray spectra, and is compared to the performances of the binned Poisson statistic (C), Pearson's χ^{2} and Neyman's χ^2_N, the KolmogorovSmirnov, and Kuiper's statistics. It is found that the unbinned log likelihood L performs best with regard to the expected chisquare distance between true and estimated spectra, the chance of a successful identification among discrete candidate models, the area under the receiveroperator 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 maximumL 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 rebinning 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 maximumL method has no lower limit on the available counts, and allows to treat weak sources which are beyond the means of binned methods.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 June 2007
 DOI:
 10.1051/00046361:20064927
 arXiv:
 arXiv:astroph/0609193
 Bibcode:
 2007A&A...468..501A
 Keywords:

 methods: statistical;
 Xrays: stars;
 stars: formation;
 Astrophysics
 EPrint:
 accepted for publication in A&