Choosing the best Rukhin goodness-of-fit statistics

The testing for goodness-of-fit in multinomial sampling contexts is usually based on the asymptotic distribution of Pearson-type chi-squared statistics. However, approximations are not justified for those cases where sample size and number of cells permit the use of adequate algorithms to calculate the exact distribution of test statistics in a reasonable time. In particular, Rukhin statistics, containing χ2 and Neyman's modified χ2 statistics, are considered for testing uniformity. Their exact distributions are calculated for different sample sizes and number of cells. Several exact power comparisons are carried out to analyse the behaviour of selected statistics. As a result of the numerical study some recommendations are given. Conclusions may be extended to testing the goodness of fit to a given absolutely continuous cumulative distribution function.