  Rejective sampling improves design and estimation efficiency of single-phase
sampling when auxiliary information in a finite population is available. When
such auxiliary information is unavailable, we propose to use two-phase
rejective sampling (TPRS), which involves measuring auxiliary variables for the
sample of units in the first phase, followed by the implementation of rejective
sampling for the outcome in the second phase. We explore the asymptotic design
properties of double expansion and regression estimators under TPRS. We show
that TPRS enhances the efficiency of the double expansion estimator, rendering
it comparable to a regression estimator. We further refine the design to
accommodate varying importance of covariates and extend it to multi-phase
sampling.
