Parameter estimation in astronomy with Poisson-distributed data.: I.: The χγ2 statistic

Applying the standard weighted mean formula, [Sigma(i) n(i) sigma(i)(-2)]/[Sigma(i)sigma(i)(-2)], to determine the weighted mean of data, n(i), drawn from a Poisson distribution, will, on average, underestimate the true mean by similar to 1 for all true mean values larger than similar to 3 when the common assumption is made that the error of the ith observation is sigma(i) = max(root n(i), 1). This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman's chi(2) statistic, chi(N)(2) = Sigma(i) (n(i) - y(i))(2)/max(n(i), 1), to compare Poisson-distributed data with model values, y(i), will typically predict a total number of counts that underestimates the true total by about 1 count per bin. Based on my finding that the weighted mean of data drawn from a Poisson distribution can be determined using the formula [Sigma(i) [n(i) + min(n(i), 1)](n(i) + 1)(-1)]/[Sigma(i)(n(i) + 1)(-1)], I propose that a new chi(2) statistic, chi(gamma)(2) = Sigma(i)[n(i) + min(n(i), 1) - y(i)](2)/[n(i) + 1], should always be used to analyze Poisson-distributed data in preference to the modified Neyman's chi(2) statistic. I demonstrate the power and usefulness of chi(gamma)(2) minimization by using two statistical fitting techniques and five chi(2) statistics to analyze simulated X-ray power-law 15 channel spectra with large and small counts per bin. I show that chi(gamma)(2) minimization with the Levenberg-Marquardt or Powell's method can produce excellent results (mean slope errors less than or similar to 3%) with spectra having as few as 25 total counts.