Improved transformation of φ-divergence goodness-of-fit test statistics based on minimum φ∗-divergence estimator for GLIM of binary data

Generalized linear models of binary data including a logistic regression model and a probit model are considered.For testing the null hypothesis that the considered model is correct, the -divergence family of goodness-offit test statistics C * that is based on a minimum * -divergence estimator is considered.The family of statistics C * includes a power divergence family of statistics R a,b that is based on a minimum power divergence estimator.The derivation of an expression of a continuous term of asymptotic expansion for the distribution of C * under the null hypothesis is shown.Using the expression, a transformed C * statistic that improves the speed of convergence to the chi-square limiting distribution of C * is obtained.In the case of R a,b , it is numerically shown that the transformed statistics usually perform better than the original statistics with respect to speed of convergence to the chi-square limiting distribution and it is also numerically shown that the power of the transformed statistics is almost the same as that of the original statistics.