Sample size allocation in two-phase sampling

When it is desired to estimate parameters of subpopulations which are not "identifiable in advance" one may use a two phase sample design. That is, one selects a large, preliminary ("first phase")sample and identifies the subpopulation to which each element belongs Thens for subpopulation j, a subsample is selectedfrom the elements identified in the first phase sample as being members of j. Finally, the variable of interest is measured for each element in this "second phase" sample In this paperweconsider both simple random and stratified random sampling at the first phase while (in each case) simple random sampling is assumed at the second phase.Two types of sample size allocation problem are investigated: (1) determination of the sample size(s) at the first phase,and (2) determination of the second phase sample sizes given the results of the first phase sample.