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D. A. van Dyk (Department of Statistics, University of California, Irvine), T. Park (Department of Statistics, Harvard University), V. L. Kashyap, A. Zezas (Harvard Smithsonian Center for Astrophysics)
Hardness ratios are commonly used to characterize the spectrum of an X-ray source when spectral fitting is not possible. However, in the case of a very small number of counts (~0 counts) or upper limits, the standard method (method of moments; based simply on the net number of counts in each band) fails to take into account the non-symmetric nature of the Poisson distribution. This results in biased estimates of the true hardness ratio. Likewise, the standard error propagation method (the delta method) does not provide realistic confidence limits. Here we present a generalized and statistically coherent scheme for computing hardness ratios and their associated errors. In this scheme we model the detected counts as a non-homogeneous Poisson process and we calculate the hardness ratios using a sophisticated Bayesian approach (taking into account local background, and effective area variations).
We apply this method on the classical hardness ratio [(H-S)/(H+S)] as well as its variants (X-ray colours [log(H/S)] and counts ratios [S/H] in different bands), and we find that in all cases the mode of the posterior hardness ratio distribution is more appropriate than its mean value (which we obtain from the standard method). Moreover we find that of all these hardness ratio variants the X-ray colours have the most symmetric posterior probability distributions. Finally we present several examples comparing this new method with the standard method, demonstrating the cases where the new method provides more accurate results.
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The author(s) of this abstract have provided an email address for comments about the abstract: dvd@ics.uci.edu
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Bulletin of the American Astronomical Society, 36 #3
© 2004. The American Astronomical Soceity.