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

Abstract

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.

Key Words:

• Chi-squared statistic
• Large sparse multinomial
• Likelihood ratio
• Limit distribution
• Power