Parameter Estimation in Astronomy with Poisson-distributed Data. I. The χγ2 Statistic

Applying the standard weighted mean formula, [ ∑iniσi-2]/[∑ iσi-2], to determine the weighted mean of data, ni, drawn from a Poisson distribution, will, on average, underestimate the true mean by ~1 for all true mean values larger than ~3 when the common assumption is made that the error of the ith observation is σi = max (√ni, 1). This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman's χ2 statistic, χN2 ≡ ∑i (ni-yi)2/max (ni, 1), to compare Poisson-distributed data with model values, yi, 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 [ ∑i [ ni + min (ni, 1)](ni + 1)-1]/[ ∑i (ni + 1)-1], I propose that a new χ2 statistic, χ2γ ≡ ∑i [ ni + min (ni, 1) - yi]-2]/[ni + 1], should always be used to analyze Poisson-distributed data in preference to the modified Neyman's χ2 statistic. I demonstrate the power and usefulness of χγ2 minimization by using two statistical fitting techniques and five χ2 statistics to analyze simulated X-ray power-law 15 channel spectra with large and small counts per bin. I show that χγ2 minimization with the Levenberg-Marquardt or Powell's method can produce excellent results (mean slope errors ≲3%) with spectra having as few as 25 total counts.