Applying the standard weighted mean formula,
![$
[\sum_i {n_i \sigma^{-2}_i}]
/
\break
[\sum_i {\sigma^{-2}_i}]
$](img1.gif)
,
to determine the weighted mean
of data,
![$n_i$](img2.gif)
, drawn from a Poisson distribution, will,
on average,
underestimate the true mean by
![$\sim$](img3.gif)
![$1$](img4.gif)
for all true mean
values larger than
![$\sim$](img3.gif)
![$3$](img5.gif)
when the common assumption is made
that the error of the
![$i$](img6.gif)
th observation is
![$\sigma_i = \max(\sqrt{n_i},1)$](img7.gif)
.
This small, but statistically significant offset,
explains the long-known observation that chi-square minimization techniques
using the modified Neyman's
![$\chi^2$](img8.gif)
statistic,
![$\chi^2_{\rm {N}} \equiv \sum_i (n_i-y_i)^2/\max(n_i,1)$](img9.gif)
,
to analyze Poisson-distributed data will
typically predict a total number of counts that
underestimates the true total
by about
![$1$](img4.gif)
count per bin.
Based on my finding that the weighted mean of data
drawn from a Poisson distribution can be
determined using the formula
![$
[
\sum_i [n_i+\min(n_i,1)](n_i+1)^{-1}
]
/
[
\sum_i (n_i+1)^{-1}
]
$](img10.gif)
, I have proposed a new
![$\chi^2$](img8.gif)
statistic,
![$\chi^2_\gamma
\equiv
\sum_i
[ n_i + \min( n_i, 1) - y_i ]^2
/
[ n_i + 1 ]$](img11.gif)
,
should always be used to analyze Poisson-distributed data
in preference to the modified Neyman's
![$\chi^2$](img8.gif)
statistic
(Mighell 1999, ApJ, 518, 380).
I demonstrated the power and usefulness of
![$\chi^2_\gamma$](img12.gif)
minimization
by using two statistical fitting techniques and three
![$\chi^2$](img8.gif)
statistics
to analyze simulated X-ray power-law 15-channel spectra
with large and small counts per bin.
I showed that
![$\chi^2_\gamma$](img12.gif)
minimization with
the Levenberg-Marquardt or Powell's method can produce
excellent results (mean errors
![${\mathrel{<\kern-1.0em\lower0.9ex\hbox{$\sim$}}}$](img13.gif)
![$3$](img5.gif)
%)
with spectra having as few as 25 total counts.