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Table of Contents - Subject Index - Author Index - PS reprint - Mighell, K. J. 2000, in ASP Conf. Ser., Vol. 216, Astronomical Data Analysis Software and Systems IX, eds. N. Manset, C. Veillet, D. Crabtree (San Francisco: ASP), 627

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

K. J. Mighell
Kitt Peak National Observatory, National Optical Astronomy Observatories, P.O. Box 26732, Tucson, AZ  85726

Abstract:

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, $n_i$, drawn from a Poisson distribution, will, on average, underestimate the true mean by $\sim$$1$ for all true mean values larger than $\sim$$3$ when the common assumption is made that the error of the $i$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 $1$ 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 ${\mathrel{<\kern-1.0em\lower0.9ex\hbox{$\sim$}}}$$3$%) with spectra having as few as 25 total counts.

© Copyright 2000 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA Next: DIRT: The Dust InfraRed Toolbox
Up: Data Analysis Tools, Techniques, and Software
Previous: Starfinder: a Code for Crowded Stellar Fields Analysis
Table of Contents - Subject Index - Author Index - PS reprint -