N. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B 46, 440-464) introduced a class of multinomial goodness-of-fit statistics R a based on power divergence. All R a have the same chi-square limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of R a under local alternatives. We obtain an expression of approximation for the distribution of R a under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of R a . We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of R a . We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance.
In a generalized linear model with binary response, the role of a link function is important to find a model that fits data well. Aranda-Ordaz (1981) proposed a family of link functions that includes a logistic link function and a complementary log-log function. In this paper, we propose a new family of models on the basis of a family of link functions by extending the family proposed by Aranda-Ordaz (1981). We also consider tests to determine whether the new model fits data well. Examples of artificial and real data showing that our new model is more appropriate than the Aranda-Ordaz model are presented.
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