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Sample Expansion for Probability Proportional to Size without Replacement Sampling

Lawrence R. Ernst


A drawback of probability proportional to size (PPS) without replacement, fixed size sampling, in comparison with equal probability sampling, is the difficulty of expanding the sample with the expanded sample remaining PPS without replacement. In fact, this author is unaware how to perform such a sample expansion for most PPS sampling schemes, such as systematic PPS sampling. However, Tille (1996) has developed a step by step PPS sampling scheme, for which at each step one unit is eliminated until the desired sample size is reached. It is shown in the current paper that sample expansion is very straightforward if Tille's method had been used to select the original sample, provided a record had been kept of the order in which units had been eliminated in selecting the original sample. Even if no such record had been kept, we demonstrate that it is still possible to expand the sample. We also discuss a modification of Tille's method that allows it attain some of the advantages of ordered systematic PPS sampling. Finally, we discuss applications to the National Compensation Survey conducted by the Bureau of Labor Statistics.