The asymptotic distribution and robustness of the likelihood ratio and score test statistics

Cordeiro & Ferrari (1991) use the asymptotic expansion of Harris (1985) for the moment generating function of the score statistic to produce a generalization of Bartlett adjustment for application to the score statistic. It is shown here that Harris's expansion is not invariant under reparameterization and an invariant expansion is derived using a method based on the expected likelihood yoke. A necessary and sufficient condition for the existence of a generalized Bartlett adjustment for an arbitrary statistic is given in terms of its moment generating function. Generalized Bartlett adjustments to the likelihood ratio and score test statistics are derived in the case where the interest parameter is one-dimensional under the assumption of a mis-specified model, where the true distribution is not assumed to be that under the null hypothesis