Bootstrapping Two-phase Sampling

Two-phase stratified sampling without replacement yields generally smaller asymptotic variance of estimators than under the convenient assumption of independence often made in practice. Motivated by the variance estimation problem, we propose and study a nonparametric bootstrap procedure for two-phase sampling. We establish conditional weak convergence of bootstrap inverse probability weighted empirical processes with several variants of calibration. Two main theoretical difficulties are the dependent observations due to sampling without replacement, and the complex limiting processes of the linear combinations of Brownian bridge processes. To address these issues, the proposed bootstrap weights take the form of the product of two weights corresponding to randomness from each phase and stratum. We apply our bootstrap to weighted likelihood estimation and establish two $Z$-theorems for a general semiparametric model where a nuisance parameter can be estimated either at a regular or a non-regular rate. We show different bootstrap calibration methods proposed in the survey sampling literature yield different bootstrap asymptotic distributions.