Worst-risk minimization in generalized structural equation models

We consider rather general structural equation models (SEMs) between a target and its covariates in several shifted environments. Given $k\in\mathbb{N}$ 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.