Generalized Linear Mixed Models (GLMMs) are widely used for analysing clustered data. One well-established method of overcoming the integral in the marginal likelihood function for GLMMs is penalized quasi-likelihood (PQL) estimation, although to date there are few asymptotic distribution results relating to PQL estimation for GLMMs in the literature. In this paper, we establish large sample results for PQL estimators of the parameters and random effects in independent-cluster GLMMs, when both the number of clusters and the cluster sizes go to infinity. This is done under two distinct regimes: conditional on the random effects (essentially treating them as fixed effects) and unconditionally (treating the random effects as random). Under the conditional regime, we show the PQL estimators are asymptotically normal around the true fixed and random effects. Unconditionally, we prove that while the estimator of the fixed effects is asymptotically normally distributed, the correct asymptotic distribution of the so-called prediction gap of the random effects may in fact be a normal scale-mixture distribution under certain relative rates of growth. A simulation study is used to verify the finite sample performance of our theoretical results.

翻译：广义线性混合模型（GLMMs）广泛应用于聚类数据分析。解决GLMMs边际似然函数中积分问题的一个成熟方法是惩罚拟似然（PQL）估计，尽管目前文献中关于GLMMs的PQL估计的渐近分布结果很少。本文针对独立聚类GLMMs，在聚类数和聚类规模均趋于无穷大的情况下，建立了参数和随机效应的PQL估计量的大样本结果。研究在两种不同框架下进行：条件于随机效应（本质上将其视为固定效应）和非条件于随机效应（将随机效应视为随机）。在条件框架下，我们证明了PQL估计量在真实固定效应和随机效应附近是渐近正态的。在非条件框架下，我们证明固定效应的估计量具有渐近正态分布，而随机效应所谓的预测间隙的正确渐近分布，在特定相对增长率下实际上可能是正态尺度混合分布。通过模拟研究验证了我们理论结果的有限样本表现。