We provide a unified approach to a method of estimation of the regression parameter in balanced linear models with a structured covariance matrix that combines a high breakdown point and bounded influence with high asymptotic efficiency at models with multivariate normal errors. Of main interest are linear mixed effects models, but our approach also includes several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. We provide sufficient conditions for the existence of the estimators and corresponding functionals, 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 high breakdown estimation with high efficiency in a wide variety of multivariate models.
翻译:我们为平衡线性模型的回归参数估算方法提供了统一的方法,配有结构化的共变矩阵,该矩阵将高分解点和约束影响结合在一起,在多变量正常差错的模型中具有高度的无症状效率。主要感兴趣的是线性混合效应模型,但我们的方法还包括若干其他标准的多变量模型,如多重回归、多变量回归、多变量位置和散射。我们为估算符和相应的函数的存在提供了充分的条件,建立了一致性和无症状正常度等无症状特性,并用分解点和影响函数来得出其稳健性。所有结果都是为一般可识别的共变形结构取得的,在对观测分布的温和条件下建立,这些结果远远超过了具有椭圆洞密度的模型。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为一般。这样,这一手稿完成并改进了高分解估计结果,在多种多变量模型中具有很高的效率。