Small-sample comparisons for the Rukhin goodness-of-fit-statistics

Rukhin's statistic family for goodness-of-fit, under the null hypothesis, has asymptotic chi-squared distribution; however, for small samples the chi-squared approximation in some cases does not well agree with the exact distribution. In this paper we consider this approximation and other three to get appropriate test levels in comparison with the exact level. Moreover, exact power comparisons for several values of the parameter under specified alternatives provide that the classical Pearson's statistic, obtained as a particular case of Rukhin statistic, can be improved by choosing other statistics from the family. An explanation is proposed in terms of the effects of individual cell frequencies on the Rukhin statistic.