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Theory

Cash (ApJ 228, 939) showed that the tex2html_wrap_inline10235 minimization criterion is a very bad one if any of the observed data bins had few counts. A better criterion is to use a likelihood function :

displaymath12971

where tex2html_wrap_inline12975 are the observed data and tex2html_wrap_inline12977 the values of the function. Minimizing C for some model gives the best-fit parameters. Furthermore, this statistic can be used in the same, familiar way as the tex2html_wrap_inline10235 statistic to find confidence intervals. One finds the parameter values that give tex2html_wrap_inline12979 , where N is the same number that gives the required confidence for the number of interesting parameters as for the tex2html_wrap_inline10235 case.

Castor (priv. comm.) has pointed out that a better function to use is :

displaymath12972

This differs from the first function by a quantity that depends only upon the data. This second function does provide a goodness-of-fit criterion similar to that of tex2html_wrap_inline10235 and it is now used in XSPEC. It is important to note that the C-statistic assumes that the error on the counts is pure Poisson, and thus it cannot deal with data that already has been background subtracted, or has systematic errors.



Keith Arnaud (kaa@genji.gsfc.nasa.gov)
Wed May 28 10:59:33 EDT 1997