  In a variety of applications, including nonparametric instrumental variable
(NPIV) analysis, proximal causal inference under unmeasured confounding, and
missing-not-at-random data with shadow variables, we are interested in
inference on a continuous linear functional (e.g., average causal effects) of
nuisance function (e.g., NPIV regression) defined by conditional moment
restrictions. These nuisance functions are generally weakly identified, in that
the conditional moment restrictions can be severely ill-posed as well as admit
multiple solutions. This is sometimes resolved by imposing strong conditions
that imply the function can be estimated at rates that make inference on the
functional possible. In this paper, we study a novel condition for the
functional to be strongly identified even when the nuisance function is not;
that is, the functional is amenable to asymptotically-normal estimation at
$\sqrt{n}$-rates. The condition implies the existence of debiasing nuisance
functions, and we propose penalized minimax estimators for both the primary and
debiasing nuisance functions. The proposed nuisance estimators can accommodate
flexible function classes, and importantly they can converge to fixed limits
determined by the penalization regardless of the identifiability of the
nuisances. We use the penalized nuisance estimators to form a debiased
estimator for the functional of interest and prove its asymptotic normality
under generic high-level conditions, which provide for asymptotically valid
confidence intervals. We also illustrate our method in a novel partially linear
proximal causal inference problem and a partially linear instrumental variable
regression problem.
