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In this chapter, a second-order correction term is considered as an index of investigating whether the distributions of statistics are close to the chi-square limiting distribution. We derive conditions for selecting a \(\phi \) -divergence statistic when considering an asymptotic test in cases where the sample size is not so large. When we use a power divergence family of statistics and the family of Rukhin’s statistics as special \(\phi \) -divergence statistics, we found that only Pearson’s \(X^2\) statistic satisfies the condition in the case of test of complete independence in a contingency table.

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The Selection of Statistics Based on a Second-Order Correction Term When Data Are Not So Large

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  • 发表日期：

    2026年

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  • Nobuhiro Taneichi,
  • Yuri Sekiya

摘要

In this chapter, a second-order correction term is considered as an index of investigating whether the distributions of statistics are close to the chi-square limiting distribution. We derive conditions for selecting a \(\phi \) -divergence statistic when considering an asymptotic test in cases where the sample size is not so large. When we use a power divergence family of statistics and the family of Rukhin’s statistics as special \(\phi \) -divergence statistics, we found that only Pearson’s \(X^2\) statistic satisfies the condition in the case of test of complete independence in a contingency table.