A two-phase sampling scheme and πps designs

A two-phase approach for sampling with unequal inclusions probabilities and fixed sample size is presented. The expansion estimator using target inclusion probabilities is suggested for estimation of population parameters. As an alternative, the estimator for two-phase sampling can be used for estimation. Inclusion probabilities are shown to be asymptotically equivalent to the targeted inclusion probabilities. By means of simulation associated estimators are shown to work well with respect to bias and precision.