Recent Advances in Parameter Estimation in Astronomy with Poisson-Distributed Data

Applying the standard weighted mean formula, $[\sum_i {n_i \sigma^{-2}_i}] / \break [\sum_i {\sigma^{-2}_i}]$ , to determine the weighted mean of data,

, drawn from a Poisson distribution, will, on average, underestimate the true mean by $\sim$

for all true mean values larger than $\sim$

when the common assumption is made that the error of the

th 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, $\chi^2_{\rm {N}} \equiv \sum_i (n_i-y_i)^2/\max(n_i,1)$ , to analyze Poisson-distributed data will typically predict a total number of counts that underestimates the true total by about

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 ${\mathrel{<\kern-1.0em\lower0.9ex\hbox{$\sim$}}}$

%) with spectra having as few as 25 total counts.

Recent Advances in Parameter Estimation in Astronomy with Poisson-Distributed Data

Abstract: