  Non-ignorable dropout is common in studies with long follow-up time, and it
can bias study results unless handled carefully. A double-sampling design
allocates additional resources to pursue a subsample of the dropouts and find
out their outcomes, which can address potential biases due to non-ignorable
dropout. It is desirable to construct semiparametric estimators for the
double-sampling design because of their robustness properties. However,
obtaining such semiparametric estimators remains a challenge due to the
requirement of the analytic form of the efficient influence function (EIF), the
derivation of which can be ad hoc and difficult for the double-sampling design.
Recent work has shown how the derivation of EIF can be made deductive and
computerizable using the functional derivative representation of the EIF in
nonparametric models. This approach, however, requires deriving the mixture of
a continuous distribution and a point mass, which can itself be challenging for
complicated problems such as the double-sampling design. We propose
semiparametric estimators for the survival probability in double-sampling
designs by generalizing the deductive and computerizable estimation approach.
In particular, we propose to build the semiparametric estimators based on a
discretized support structure, which approximates the possibly continuous
observed data distribution and circumvents the derivation of the mixture
distribution. Our approach is deductive in the sense that it is expected to
produce semiparametric locally efficient estimators within finite steps without
knowledge of the EIF. We apply the proposed estimators to estimating the
mortality rate in a double-sampling design component of the President's
Emergency Plan for AIDS Relief (PEPFAR) program. We evaluate the impact of
double-sampling selection criteria on the mortality rate estimates.
