Selection of statistics for a multinomial goodness-of-fit test and a test of independence for a multi-way contingency table when data are sparse

For the goodness-of-fit test for a multinomial distribution and test of some kinds of independence of a multi-way contingency table, we consider test statistics based on the \(\phi \) $\varphi$ -divergence family. Members of the \(\phi \) $\varphi$ -divergence family of statistics all have an equivalent Chi-square limiting distribution under the null hypothesis. We consider a second-order correction term as an index of investigating whether the distributions of statistics are close to the Chi-square limiting distribution. We derive properties for the second-order correction term for selecting a \(\phi \) $\varphi$ -divergence statistic when we consider an asymptotic test in the case of data being sparse. We propose a selection of statistics when we use a power divergence family of statistics and the family of Rukhin’s statistics as special \(\phi \) $\varphi$ -divergence statistics.