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Allocating Sample to Strata Proportional to Aggregate Measure of Size with Both Upper and Lower Bounds on the Number of Units in Each Stratum

Lawrence R. Ernst and Christopher J. Guciardo


For stratified PPS sampling, it is common to allocate the sample among strata proportional to their aggregate measures of size. In some cases, this allocation yields some strata whose sample sizes exceed their frame sizes, in which case the sample sizes for these strata are reduced to their frame sizes, and the remaining sample units are allocated among the remaining strata proportional to their aggregate measures of size. The process is repeated as necessary. An analogous iterative procedure may be used if there are lower bounds on the number of sample units in each stratum and these constraints are violated, but the sample size does not exceed the frame size for any strata. However, as explained in the paper, when the proportional allocation produces both strata with sample sizes above upper bounds and strata with sizes below lower bounds, an algorithm that sets the sample sizes for all violating strata to their respective bounds does not lead to an optimal allocation. An algorithm is presented that does yield an optimal allocation. An application to the sample selection for the National Compensation Survey is discussed.