Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model |
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Authors: | Nobuhiro Taneichi Yuri Sekiya |
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Affiliation: | 1. Department of Mathematics and Computer Science , Kagoshima University , Kagoshima, Japan taneichi@sci.kagoshima-u.ac.jp;3. Hokkaido University of Education , Kushiro, Japan |
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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. |
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Keywords: | Chi-square distribution Edgeworth expansion Moment correction Null distribution Product multinomial model Test of homogeneity |
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