Exponential consistency of M-estimators in generalized linear mixed models

Pub Date : 2024-08-08 DOI:10.1016/j.jspi.2024.106222
Andrea Bratsberg , Magne Thoresen , Abhik Ghosh
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Abstract

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 has 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.

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广义线性混合模型中 M 估计器的指数一致性
广义线性混合模型是分析聚类数据的强大工具,其中的未知参数通常(也是最常用的)通过最大似然和限制最大似然程序进行估计。然而,众所周知,基于似然法的程序对异常值非常敏感,因此,M-估计器作为一种在可能的数据污染情况下获得稳健估计值的方法而备受青睐。本文证明,对于定义广义线性混合模型中 M-estimators 的足够平滑的一般损失函数,估计值与真实回归系数之间偏差的尾部概率具有指数约束。这意味着在适当的假设条件下,这些 M-estimators 的指数一致性率,将现有的指数一致性结果从单变量推广到多变量响应。我们在线性模型和逻辑混合模型中,以最大似然估计器和稳健最小密度功率发散估计器(基于模型的 M-estimators 的一个流行例子)为例,进一步说明了这一理论结果,并通过模拟研究将其与经验收敛率进行了比较。
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