Chi-square statistic with the Gehrels variance function.
chi2gehrels is the default statistic in Sherpa.
If the number of counts in each bin is small (< 5), then we
cannot assume that the Poisson distribution from which the
counts are sampled has a nearly Gaussian shape. The standard
deviation (i.e., the square-root of the variance) for this
low-count case has been derived by Gehrels (1986):
sigma(i,S) = 1 + (sqrt)[N(i,S)+0.75] .
Higher-order terms have been dropped from the expression; it
is accurate to approximately one percent. If one does not
perform background subtraction, then sigma(i) = sigma(i,S); otherwise, one may
use standard error propagation to estimate that
sigma(i)^2 = sigma(i,S)^2 + [A(S)/A(B)]^2 sigma(i,B)^2 .
The background term appears only if a background region is
specified and background subtraction is done. The help file
on the chi-square statistic has more information, including
definitions of the quantities in the equation:
"ahelp chisquare"
.
The accuracy of the latter expression has not been determined,
thus the user should proceed with caution when subtracting
background from the raw data when using this statistic. An
approach preferable to background subtraction is to model the
background and data simultaneously.
sherpa> set_stat("chi2gehrels");
sherpa> show_stat();
Statistic: Chi2Gehrels
Set the fitting statistic and then confirm the new value.
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