Semiparametric Efficient Inference under Non-Informative Complex Survey Designs

Two features intrinsic to survey sampling complicate semiparametric efficiency analysis: design-induced dependence among sampling indicators and the randomness of finite-population targets under the superpopulation law. For general semiparametric full-data models, we show that the observed-data experiment under a broad class of dependent designs is locally asymptotically normal with the tangent-space structure of a reference Poisson experiment. Under the joint superpopulation-design law, first-order efficiency depends on the design only through the limiting inclusion-probability function. Standard missing-at-random projection in the reference experiment characterizes the observed-data efficient influence function. Finite-population targets are treated through first-order asymptotic expansions, extending the analysis beyond exact random sums to nonlinear census characteristics. Superpopulation and finite-population centerings yield equivalent notions of local regularity and efficiency, with their bounds linked by a Pythagorean decomposition that gives a generalized finite-population correction. We then give general and design-specific conditions under which cross-fitted estimators with estimated nuisance functions attain both efficiency bounds. For the finite-population mean, the bound equals the large-sample limit of the Godambe-Joshi anticipated-variance lower bound. For scalar targets, we characterize optimal limiting inclusion probabilities. Simulations and California Academic Performance Index data illustrate the theory.