  Approximations to the modified signed likelihood ratio statistic are
asymptotically standard normal with error of order $n^{-1}$, where $n$ is the
sample size. Proofs of this fact generally require that the sufficient
statistic of the model be written as $(\hat{\theta},a)$, where $\hat{\theta}$
is the maximum likelihood estimator of the parameter $\theta$ of the model and
$a$ is an ancillary statistic. This condition is very difficult or impossible
to verify for many models. However, calculation of the statistics themselves
does not require this condition. The goal of this paper is to provide
conditions under which these statistics are asymptotically normally distributed
to order $n^{-1}$ without making any assumption about the sufficient statistic
of the model.
