Parameter Estimation in Astronomy with Poisson - Distributed Data . I . the Χ 2 Γ Statistic

, 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 σi = max( √ ni, 1). This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman’s χ statistic, χ2N ≡ ∑ i(ni − yi)/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