  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.
