Second term improvement to generalized linear mixed model asymptotics

A recent article by Jiang et al. (2022) on generalized linear mixed model asymptotics 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.