Improved inference in dispersion models

We derive a general matrix Bartlett-type correction factor to the gradient statistic in the class of dispersion models. The correction improves the large-sample χ2 approximation to the null distribution of the gradient statistic when the sample size is finite. We conduct Monte Carlo simulation experiments to evaluate and compare the performance of various different tests, namely the usual Wald, likelihood ratio, score, and gradient tests, the Bartlett-corrected versions of the likelihood ratio, score, and gradient tests, and bootstrap-based tests. The simulation results suggest that the analytical and computational corrections are effective in removing size distortions of the type I error probability with no power loss. The impact of the corrections in two real data applications is considered for illustrative purposes.