We provide a unified approach to S-estimation in balanced linear models with structured covariance matrices. Of main interest are S-estimators for linear mixed effects models, but our approach also includes S-estimators in several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. We provide sufficient conditions for the existence of S-functionals and S-estimators, establish asymptotic properties such as consistency and asymptotic normality, and derive their robustness properties in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some of our results are new and others are more general than existing ones in the literature. In this way this manuscript completes and improves results on S-estimation in a wide variety of multivariate models. We illustrate our results by means of a simulation study and an application to data from a trial on the treatment of lead-exposed children.

中文翻译：

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具有结构化协方差矩阵的线性模型中的 S 估计

我们在具有结构化协方差矩阵的平衡线性模型中提供了一种统一的 S 估计方法。主要感兴趣的是线性混合效应模型的 S 估计器，但我们的方法还包括其他几个标准多元模型中的 S 估计器，例如多元回归、多元回归和多元位置和散布。我们为S-泛函和S-估计量的存在提供了充分条件，建立了一致性和渐近正态性等渐近性质，并推导出了它们在击穿点和影响函数方面的鲁棒性。所有结果都是针对一般可识别的协方差结构获得的，并且是在观察分布的温和条件下建立的，这远远超出了具有椭圆轮廓密度的模型。我们的一些结果是新的，而另一些则比文献中的现有结果更普遍。通过这种方式，本手稿完成并改进了各种多元模型中 S 估计的结果。我们通过模拟研究和对铅暴露儿童治疗试验数据的应用来说明我们的结果。