Improved inference in dispersion models

Abstract

• A Bartlett-type correction to the gradient statistic in dispersion models is derived. • The correction improves the chi-squared approximation when the sample size is finite. • A comprehensive simulation study compares several different tests in small samples. • Large sample tests, including the gradient test, may perform poorly in small samples. • Bartlett and Bartlett-type corrections remove size distortions with no power loss. 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 χ 2 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.