Worst-risk minimization in generalized structural equation models

Abstract We consider extended structural equation models, whereby a target of interest and its covariates are considered in several shifted environments. Given $k\in{\mathbb{N}}$ shift environments, we consider the collection of all shifts that are at most $\gamma $-times as strong as a given weighted linear combination of these $k$ shifts together with its associated worst (quadratic) risk. This worst risk has a convenient decomposition with an explicit population minimizer. We consider its corresponding plug-in estimator. We show that this plug-in estimator is (almost surely) consistent and satisfies a concentration in measure result. The solution to the worst risk minimizer is rather reminiscent of the corresponding ordinary least squares solution in that it is a 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 causal inference or anchor regression but the concentration in measure result and the moment bounds are new even in the linear setting.