ABSTRACT
Many large sample surveys are analyzed as though the sampling was done with replacement, even if a without-replacement sample was collected, because the estimators of the variance for with-replacement samples have simple form and require less information. Many large complex surveys are so highly stratified that each stratum contains only a few primary sampling units (psus). A large number of strata are used to increase the precision of the survey estimates. The idea behind unequal-probability sampling is simple. There are several ways to sample psus with unequal probabilities. Systematic sampling is often used to select psus in large complex samples, rather than generating random numbers with replacement. Systematic sampling usually gives a sample without replacement, but in large populations sampling without replacement and sampling with replacement are very similar, as the probability that a unit will be selected twice is small. The estimators for two-stage unequal-probability sampling with replacement are almost the same as those for one-stage sampling.
