Stratified sampling using double samples

Usual stratified sampling design assume that one is able to draw units directly from given strata. If this is not possible, one can use the following double sampling procedure: First take a large simple random sample out of the whole population and find out, to which stratum each sample unit belongs. Out of these chosen units take a second stratified sample. In this paper unbiased estimators for this procedure in the cases of known (part I) and unknown (part II) stratum weights are proposed for sampling with replacement and sampling without replacement and their variances are evaluated.