We consider rather general structural equation models (SEMs) between a target and its covariates in several shifted environments. Given $k\in\N$ shifts we consider the set of shifts that are at most $\gamma$-times as strong as a given weighted linear combination of these $k$ shifts and the worst (quadratic) risk over this entire space. This worst risk has a nice decomposition which we refer to as the "worst risk decomposition". Then we find an explicit arg-min solution that minimizes the worst risk and consider its corresponding plug-in estimator which is the main object of this paper. This plug-in estimator is (almost surely) consistent and we first prove a concentration in measure result for it. The solution to the worst risk minimizer is rather reminiscent of the corresponding ordinary least squares solution in that it is product of a vector and an inverse of a Grammian matrix. Due to this, the central moments of the plug-in estimator is not well-defined in general, but we instead consider these moments conditioned on the Grammian inverse being bounded by some given constant. We also study conditional variance of the estimator with respect to a natural filtration for the incoming data. Similarly we consider the conditional covariance matrix with respect to this filtration and prove a bound for the determinant of this matrix. This SEM model generalizes the linear models that have been studied previously for instance in the setting of casual inference or anchor regression but the concentration in measure result and the moment bounds are new even in the linear setting.

翻译：我们考虑了目标变量与其协变量在多个偏移环境下的相当一般的结构方程模型(SEM)。给定$k\in\N$个偏移，我们考虑这些$k$个偏移的加权线性组合中强度不超过$\gamma$倍的偏移集合，以及整个空间上的最坏（二次）风险。该最坏风险具有一个良好的分解，我们称之为“最坏风险分解”。随后，我们找到了最小化该最坏风险的显式arg-min解，并考虑其相应的插件估计量，这是本文的主要研究对象。该插件估计量是（几乎必然）一致的，我们首先证明了它的一个测度集中结果。最坏风险最小化问题的解在形式上类似于普通最小二乘解，即表现为一个向量与一个Gram矩阵逆的乘积。因此，该插件估计量的中心矩通常定义不充分，但我们转而考虑当Gram矩阵逆被某个给定常数界定时这些矩的条件形式。我们还研究了关于输入数据自然滤子的估计量条件方差。类似地，我们考虑了关于该滤子的条件协方差矩阵，并证明了该矩阵行列式的一个界。该SEM模型推广了此前在因果推断或锚点回归等场景中研究的线性模型，但测度集中结果和矩界即使在线性设定下也是全新的。