In order to estimate the mean of a finite population from a random sample, the sample units may be selected in two ways. A pre-determined number $m$ of units may be selected with replacement or sampling with replacement may be continued till a desired number $n$ of distinct units is obtained. The first procedure is called sampling with replacement while the second may be called sampling without replacement. A comparison is generally made between the two procedures with $m = n.$ This comparison, however, is not fair since costs are usually proportional to the number of distinct units and in the first procedure this number would be less than or equal to $n.$ In the first procedure the population mean is usually estimated by the sample mean based on all the units in the sample including repetitions while in the second procedure the estimate is generally made to depend on the distinct units only. The object of this paper is to show that the estimate making use of only the distinct units is superior in either procedure. The following results are proved in this paper: (i) In sampling with replacement the estimate of the mean based on distinct units in the sample is superior to the estimate based on the total sample size when (a) the total sample size is fixed in advance, while the number of distinct units in the sample is a random variable, and, (b) when the total sample size is a random variable while the number of distinct units is fixed in advance. (ii) The same is true of ratio estimates. It is also shown that the bias is numerically less if the ratio estimate is based on distinct units regardless of whether these are fixed in advance or considered as random variables. (iii) Expressions for the estimation of the variances of the various estimates considered in this paper are given. (iv) The above results are extended to multistage sampling.
"Some Remarks on Sampling with Replacement." Ann. Math. Statist. 29 (2) 550 - 557, June, 1958. https://doi.org/10.1214/aoms/1177706630