Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model

Abstract

Statistics R ^a based on power divergence can be used for testing the homogeneity of a product multinomial model. All R ^a have the same chi-square limiting distribution under the null hypothesis of homogeneity. R ⁰ is the log likelihood ratio statistic and R ¹ is Pearson's X ² statistic. In this article, we consider improvement of approximation of the distribution of R ^a under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R ^a under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R ^a is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R ^a when a ≤ 0, which include the log likelihood ratio statistic.

Keywords:

• Chi-square distribution
• Edgeworth expansion
• Moment correction
• Null distribution
• Product multinomial model
• Test of homogeneity

Mathematics Subject Classification:

• Primary 62E20
• Secondary 62H10

Acknowledgment

The authors are very grateful to a reviewer for valuable comments and suggestions. The research was supported by Grant-in-Aid for Scientific Research, No. 16500168, Japan Society for the Promotion of Science.