A power approximation of the test of homogeneity for multinomial populations based on a normalizing transformation

Abstract

Cressie and Read(1984) introduced multino~riial goodness-of-fit statistics based on a, class of divergence measures I_a between discrete distributions. It was also introduced that a class of statistics R_a for the test of homogeneity for multino~riial populations based on I_a (Read and Cressie(1988)). This class includes Pearson's X₂ statistic (when a = 1) and the log-likelihood ratio statistic (when a = 0). All R_a have the same chi-squared limiting null distribution. The power of the class is ordinary approximated from a noncentral chi-squared distribution that is also the same for all a. Applying the power approximation theories for the multinomial goodness-of-fit test developed by Broffitt and Randles(1977) and Drost et al.(l989), Taneichi and Sekiya(1995) proposed three approximations to the power of R_a that vary with the statistic chosen. In this paper we propose a. new approximation to the power of R_a The new approximation is a normal approximation based on a normalizing transformation of the statistics. Moreover, the proposed approximation and the other approximations are numerically compared with the power directly calculated from the product multinomial model. As a result of comparison, the new approximation is found to be effective for the statistics when a a≤0.

Keywords:

• Noncentral chi-squared distribution
• Normalizing transfoima-tion
• Product multinomial model
• Test of homogeneity