  A recent article on generalised linear mixed model asymptotics, Jiang et al.
(2022), derived the rates of convergence for the asymptotic variances of
maximum likelihood estimators. If $m$ denotes the number of groups and $n$ is
the average within-group sample size then the asymptotic variances have orders
$m^{-1}$ and $(mn)^{-1}$, depending on the parameter. We extend this theory to
provide explicit forms of the $(mn)^{-1}$ second terms of the asymptotically
harder-to-estimate parameters. Improved accuracy of studentised confidence
intervals is one consequence of our theory.
