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

Five different statistics are compared with respect to parameter, error, an d goodness-of-st estimation in the case of counting experiments. In particu lar, 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 esti mate parameters. If goodness-of-fit estimations are to be done, Pearson's c hi-square should be used. It is the only statistic that leads to the correc t 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 o f 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 und er a peak, Poisson MLE was shown to be the only statistic that comes to con sistent and unbiased results, two other statistics give asymptotically cons istent results. The widely used Neyman's chi-square fails in all cases. Fur ther, 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 sti ll can be used. Special emphasis is put on the evaluation of Doppler-broade ned gamma line shapes as they are measured in the Crystal-GRID technique. ( C) 2001 Elsevier Science B.V. All rights reserved.