Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However, since the likelihood based procedures are known to be highly sensitive to outliers, M-estimators have become popular as a means to obtain robust estimates under possible data contamination. In this paper, we prove that, for sufficiently smooth general loss functions defining the M-estimators in generalized linear mixed models, the tail probability of the deviation between the estimated and the true regression coefficients have an exponential bound. This implies an exponential rate of consistency of these M-estimators under appropriate assumptions, generalizing the existing exponential consistency results from univariate to multivariate responses. We have illustrated this theoretical result further for the special examples of the maximum likelihood estimator and the robust minimum density power divergence estimator, a popular example of model-based M-estimators, in the settings of linear and logistic mixed models, comparing it with the empirical rate of convergence through simulation studies.

翻译：广义线性混合模型是分析聚类数据的强大工具，其中未知参数通常（且最常见地）通过最大似然和限制最大似然方法进行估计。然而，由于基于似然的方法对异常值高度敏感，M估计量作为一种在可能存在数据污染的情况下获得稳健估计的手段已变得流行。本文证明，对于定义广义线性混合模型中M估计量的充分光滑的一般损失函数，估计的回归系数与真实回归系数之间偏差的尾部概率具有指数界。这暗示了在适当假设下这些M估计量的指数一致性速率，将现有的指数一致性结果从单变量响应推广到多变量响应。我们进一步通过线性混合模型和逻辑混合模型中的特例——最大似然估计量和稳健的最小密度功率散度估计量（一种基于模型的M估计量的流行例子）——阐述了这一理论结果，并通过模拟研究将其与经验收敛速率进行了比较。