Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments

Five different statistics are compared with respect to parameter, error, and goodness-of-fit estimation in the case of counting experiments. In particular, maximum likelihood approaches are opposed to chi-square techniques. It could be shown that the maximum likelihood estimation derived for Poisson distributed data (Poisson MLE) produces the best statistic in order to estimate parameters. If goodness-of-fit estimations are to be done, Pearsonʹs chi-square should be used. It is the only statistic that leads to the correct expectation value for chi-square. All the other statistics do not follow a chi-square distribution. It is discussed that the chi-square per degree of freedom is not well suited for judging the consistency of a model and the data. When estimating the mean of Poisson distributed data or the area under a peak, Poisson MLE was shown to be the only statistic that comes to consistent and unbiased results, two other statistics give asymptotically consistent results. The widely used Neymanʹs chi-square fails in all cases. Further, artificial Poisson distributed data have been created on the basis of known model functions. It is shown and discussed in which cases chi-square techniques fail to extract the correct parameter values and where they still can be used. Special emphasis is put on the evaluation of Doppler-broadened gamma line shapes as they are measured in the Crystal-GRID technique.