Improvement of test in contingency tables based on asymptotic approximation.

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1. In the case that r X s contingency table is considered as full multinomial model, we derive expression of asymptotic expansion for the distribution of the test statistics based on Φ-divergence under independent null hypothesis. The expression consists of continuous and discontinuous terms. Evaluation for the continuous and discontinuous term is also considered. Using the term of asymptotic expansion assuming a continuous distribution in the expression, we construct transformation for improving small-sample accuracy of chi-square approximation of the distribution of the statistic under independent hypothesis.2. In the case that r X s contingency table is considered as product multinomial model, we derive expression of asymptotic expansion for the distribution of the test statistics based on power-divergence under null hypothesis that all of the multinomial populations are homogeneity. The expression consists of continuous and discontinuous terms. Evaluation for the continuous and discontinuous term is also considered when the number of samples from each population is the same.3. In the test of independence in full multinomial model, by assuming the continuity of the random variables, we derive asymptotic expansions for the distribution of the test statistics based on power-divergence under local alternative and non-local alternative hypotheses, respectively.