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Paper: Recent Advances in Parameter Estimation in Astronomy with Poisson-Distributed Data
Volume: 216, Astronomical Data Analysis Software and Systems IX
Page: 627
Authors: Mighell, K. J.
Abstract: Applying the standard weighted mean formula, [sum_i {n_i sigma(-2}_i) ]/[sum_i {sigma(-2}_i) ], to determine the weighted mean of data, n_i, 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 sigma_i = max(sqrt{n_i},1). This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques using the modified Neyman's chi(2) statistic, chi(2N) equiv sum_i (n_i-y_i)(2/max(n_i,1)) , to analyze Poisson-distributed data 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 [sum_i [n_i+min(n_i,1)](n_i+1)(-1) ]/[sum_i (n_i+1)(-1) ], I have proposed a new chi(2) statistic, chi(2_gamma) equiv sum_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 (Mighell 1999, ApJ, 518, 380). I demonstrated the power and usefulness of chi(2_gamma) minimization by using two statistical fitting techniques and three chi(2) statistics to analyze simulated X-ray power-law 15-channel spectra with large and small counts per bin. I showed that chi(2_gamma) minimization with the Levenberg-Marquardt or Powell's method can produce excellent results (mean errors {mathrel{<kern-1.0emlower0.9exhbox∼}}3%) with spectra having as few as 25 total counts.
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