  Suppose we are interested in the mean of an outcome variable missing not at
random. Suppose however that one has available a fully observed shadow
variable, which is associated with the outcome but independent of the
missingness process conditional on covariates and the possibly unobserved
outcome. Such a variable may be a proxy or a mismeasured version of the outcome
available for all individuals. We have previously established necessary and
sufficient conditions for identification of the full data law in such a
setting, and have described semiparametric estimators including a doubly robust
estimator of the outcome mean. Here, we propose two alternative doubly robust
estimators for the outcome mean, which may be viewed as extensions of analogous
methods under missingness at random, but enjoy different properties. We assess
correctness of the required working models via straightforward goodness-of-fit
tests.
