# Deductive semiparametric estimation in Double-Sampling Designs with application to PEPFAR

## TL;DR

This paper develops a deductive, semiparametric estimation method for double-sampling designs, enabling robust survival probability estimation without complex influence function derivations, demonstrated through application to PEPFAR mortality data.

## Contribution

It introduces a discretized support-based approach for semiparametric estimation in double-sampling designs, simplifying the derivation process and ensuring local efficiency.

## Key findings

- Proposed estimators effectively estimate mortality rates in PEPFAR data.
- Discretized support approach approximates complex distributions successfully.
- Method reduces reliance on difficult influence function derivations.

## Abstract

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.