Parameter Estimation in Astronomy with Poisson-Distributed Data. I. The Chi-Square-Gamma Statistic
Abstract
Applying the standard weighted mean formula, [sumi ni sigma-2i] / [sumi sigma-2i], 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 sigmai = max(sqrtni,1). This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman's chi-square statistic, chi2N equiv sumi (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 [sumi [ni + min(ni,1)] (ni+1)-1] / [sumi (ni+1)-1], I propose that a new chi-square statistic, chi2gamma equiv sumi [ni + min(ni,1) - yi]2 / [ni + 1], should always be used to analyze Poisson-distributed data in preference to the modified Neyman's chi-square statistic. I demonstrate the power and usefulness of chi-square-gamma minimization by using two statistical fitting techniques and five chi-square statistics to analyze simulated X-ray power-law 15-channel spectra with large and small counts per bin. I show that chi-square-gamma 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.
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