Hypothesis testing for meagre data sets

Improved chi(2) statistics are defined for both Poisson and multinomial data sets. Numerical experiments with the Nousek-Shue test problem from X-ray spectroscopy indicate that, in contrast to Pearson's statistic X-2, these modified statistics remain approximately valid at the 2 sigma level of significance even when the mean counts per bin is much less than 1 provided that the total counts N greater than or similar to 30. The simplest of the proposed statistics is formally equivalent to carrying out a goodness-of-fit test with Pearson's statistic but with modified confidence limits, and this procedure is recommended.