Second term improvement to generalised linear mixed model asymptotics

Summary 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 statistical inference and planning are consequences of our theory.