Cash (ApJ **228**, 939) showed that the 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 :

where

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

This differs from the first function by a quantity that depends only upon the data. In the limit of a large number of counts this second function does provide a goodness-of-fit criterion similar to that of 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.

Arnaud (2001, ApJ submitted) has extended the method of Cash to include the case when a background spectrum is also in use. Note that this requires the source and background spectra to both be available, it does not work on a background-subtracted spectrum.

Suppose we have an observation which produces *S*_{i} events in the
spectral bins in an exposure time of *t*_{s}. This
observation includes events from the source of interest along with
background events. Further suppose that we perform a background
observation which generates *B*_{i} events in an exposure time *t*_{b}.
If the model source count rate in bin *i* is *y*_{i} then the new fit
statistic is

where

and

In the limit of large numbers of counts/bin a second-order Taylor expansion shows that W tends to

which is distributed as with (