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Summary It is shown that sampling without replacement with probability proportional to size can be achieved if the units are grouped with reference to size. When the same unit is chosen a second time, it is substituted by another unit of the same size chosen at random. The estimate of the population total is formally the same as when sampling is done with replacement. The estimate of variance differs in that from the sum of squares of deviations of the ratio, r, we subtract, for each group chosen t ( 1) times, the quantity tS/N, where S is the sum of squares of deviations within the group and N the number in the group.
Wallace Stevens (Tue,) studied this question.