Improved Approximations for the Distributions of Multinomial Goodness-of-fit Statistics Based on φ-divergence under Nonlocal Alternatives

Zografos et al. (1990) introduced the φ-divergence family of statistics Cφ to the goodness-of-fit test. The φ-divergence family of statistics Cφ includes the power divergence family of statistics proposed by Cressie and Read (Cressie and Read (1984) and Read and Cressie (1988)) as a special case. Sekiya and Taneichi (2004) derived the multivariate Edgeworth expansion assuming a continuous distribution for the distributions of power divergence statistics under a nonlocal alternative hypothesis. In this paper, we consider an expansion for the family of general φ-divergence statistics Cφ. We derive the multivariate Edgeworth expansion assuming a continuous distribution for the distribution of Cφ under a nonlocal alternative hypothesis. By using the expansion, we propose a new approximation for the power of the statistic Cφ. We numerically investigate the accuracy of the approximation when two types of concrete φ-divergence statistics are applied. By the numerical investigation, we show that the present approximation is a good approximation especially when alternative hypotheses are distant from the null hypothesis.