  We consider identification and inference about mean functionals of observed
covariates and an outcome variable subject to nonignorable missingness. By
leveraging a shadow variable, we establish a necessary and sufficient condition
for identification of the mean functional even if the full data distribution is
not identified. We further characterize a necessary condition for
$\sqrt{n}$-estimability of the mean functional. This condition naturally
strengthens the identifying condition, and it requires the existence of a
function as a solution to a representer equation that connects the shadow
variable to the mean functional. Solutions to the representer equation may not
be unique, which presents substantial challenges for nonparametric estimation
and standard theories for nonparametric sieve estimators are not applicable
here. We construct a consistent estimator for the solution set and then adapt
the theory of extremum estimators to find from the estimated set a consistent
estimator for an appropriately chosen solution. The estimator is asymptotically
normal, locally efficient and attains the semiparametric efficiency bound under
certain regularity conditions. We illustrate the proposed approach via
simulations and a real data application on home pricing.
