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

Abstract

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 $\varphi$-divergence family of goodness-of-fit test statistics $C_{\varphi {\varphi }^{*}}$ that is based on a minimum ${\varphi }^{*}$-divergence estimator is considered. The family of statistics $C_{\varphi {\varphi }^{*}}$ 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 $C_{\varphi {\varphi }^{*}}$ under the null hypothesis is shown. Using the expression, a transformed $C_{\varphi {\varphi }^{*}}$ statistic that improves the speed of convergence to the chi-square limiting distribution of $C_{\varphi {\varphi }^{*}}$ 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.