The copyright of this article belongs to the Japan Statistical Society. Permission to reuse the material has been obtained from the Japan Statistical Society. Since we consider a family of test statistics based on \(\phi \) -divergence as a generalized goodness-of-fit test statistics for multinomial model throughout this book, first we introduce the test statistic for a simple null hypothesis and a composite null hypothesis. Special families of \(\phi \) -divergence statistics are also introduced. Second, we show the history of development and summarize the approximation of distribution of a multinomial goodness-of-fit test statistic based on an asymptotic expansion. Third, we introduce the transformation of test statistics which improve the accuracy of the chi-square approximation of a test statistic by using the expression of an asymptotic expansion of the distribution of the test statistic.