Parameter Estimation in Astronomy with Poisson‐distributed Data. I.The \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \…

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 froma Poisson distribution, will, on average, underestimate the true mean by ~1 forall true mean values larger than ~3 when the common assumption is made that theerror of the ith observation is sigma_i = max(sqrt{n_i},1). This small, butstatistically significant offset, explains the long-known observation thatchi-square minimization techniques which use the modified Neyman's chi-squarestatistic, chi^2_{N} equiv sum_i (n_i-y_i)^2 / max(n_i,1), to comparePoisson-distributed data with model values, y_i, will typically predict a totalnumber 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 Poissondistribution 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 propose that a new chi-squarestatistic, 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 themodified Neyman's chi-square statistic. I demonstrate the power and usefulnessof chi-square-gamma minimization by using two statistical fitting techniquesand five chi-square statistics to analyze simulated X-ray power-law 15-channelspectra with large and small counts per bin. I show that chi-square-gammaminimization with the Levenberg-Marquardt or Powell's method can produceexcellent results (mean slope errors <=3%) with spectra having as few as 25total counts.