Small-Sample Comparisons for the Power Divergence Goodness-of-Fit Statistics

The small-sample properties of the power divergence family of goodness-of-fit statistics (Cressie and Read in press) are examined. The asymptotic χ2 and three alternative approximations are used to obtain appropriate test levels for comparison with the exact level. It is shown that if the χ2 approximation is to be used, then there are many preferable statistics to the log-likelihood ratio G 2 (including Pearson's χ2). Furthermore, exact power comparisons under specific alternatives to the symmetric null hypothesis show that the power of the well-known tests based on G 2 and χ2 can be improved by choosing other statistics from the family. An explanation is proposed in terms of the effect of relative deviations between observed and expected counts.