A new variance estimator for a two-phase restratified sample with PPS sampling at both phases

In this paper we consider variance estimation for population totals and ratios in complex, cross-stratified surveys in which the ultimate sampling weights are random variables, dependent on the first phase of sampling. A new hybrid variance estimator, dependent on both model-based and design-based ideas, is introduced. Theoretical and empirical justifications are given which demonstrate that the proposed method handles well the difficult aspects of this sample design.