Efficient Transported Distributional and Quantile Treatment Effects with Surrogate-Assisted Missing Primary Outcomes

We study target-population distributional and quantile treatment effects when a source study observes treatment and post-treatment surrogates for all source units but observes a long-run primary outcome only for a validation subset, while the target population contributes only baseline covariates. The target estimands are transported counterfactual distribution functions $\psi_a(y)=P(Y^a\le y\mid R=0)$, their quantiles $q_a(\tau)$, and the quantile treatment effect $\Delta(\tau)=q_1(\tau)-q_0(\tau)$. The surrogate is not treated as a replacement endpoint and no Prentice-type surrogacy condition is imposed. Instead, the surrogate is used only to improve efficiency under missing-at-random primary-outcome sampling. We derive the nonparametric efficient influence function, which has three orthogonal components corresponding to target covariate sampling, the source surrogate process, and missing primary outcomes. This yields a closed-form cross-fitted one-step estimator after nuisance estimation. We establish identification, the canonical gradient, exact drift identities, ratio-level robustness, pointwise and uniform asymptotic linearity for transported CDFs, Bahadur representations for quantiles under explicit local inverse-map conditions, high-level multiplier-bootstrap simultaneous bands under explicit estimated-process and density conditions, and quantile-specific efficiency gains from observing surrogates. We also give lower-level nuisance-rate verification for a deliberately restricted class of analyzable bounded finite-dimensional or finite-rank implementations based on sieve ridge regression, ridge logistic regression, calibrated density-ratio estimation, finite-rank kernel ridge regression, and isotonic projection under explicit grid, eigenvalue, source, and entropy conditions.