  We propose a nonparametric bootstrap procedure for two-phase stratified
sampling without replacement. In this design, a weighted likelihood estimator
is known to have smaller asymptotic variance than under the convenient
assumption of independence often made in practice. Variance estimation,
however, has not been well studied for semiparametric models where variance may
not have a closed form. Motivated by semiparametric inference, we establish
conditional weak convergence of bootstrap inverse probability weighted
empirical processes with several variants of calibration. Two main obstacles to
applying existing bootstrap empirical process theory are the dependent and
biased sample due to sampling design, 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.
