Selection without replacement from within strata with probability proportional to size

In selection with probability proportional to size x from within strata without replacement, the usual method of selection gives rise to bias in the estimate of the total of a variate y derived by weighting the units by weights proportional to 1/x. By means of numerical examples it is shown that the amount of this bias is usually quite trivial. If, however, unbiased estimates are required, the true total probabilities of selection of the different units can be calculated easily for samples of 2, and with considerably more labour for samples of 3. The bias in the ordinary formula for the estimation of error is also investigated, and the formula is shown to be reasonably accurate. Horvitz and Thompson have given an unbiased estimator of the error variance, but this is shown to be inefficient and a new unbiased estimator is given. A method of revising the size measures so that with the usual method of selection the true total probabilities of selection are proportional to the original size measures is given for samples of 2. Horvitz and Thompson's solution of this problem does not appear to give satisfactory approximations in the cases met with in practice. The selection of successive members of a sample with arbitrary sets of probabilities chosen solely so that the total probabilities shall be proportional to the original size measures, which has been advocated in various quarters, is criticized.