Sampling with Unequal Probabilities and without Replacement

Abstract

Given a population of $N$ units, it is required to draw a random sample of $n$ distinct units in such a way that the probability for the $i$th unit to be in the sample is proportional to its "size" $x_i$ (sampling with p.p.s. without replacement). From a number of alternatives of achieving this, one well known procedure is here selected: The $N$ units in the population are listed in a random order and their $x_i$ are cumulated and a systematic selection of $n$ elements from a "random start" is then made on the cumulation. The mathematical theory associated with this procedure, not available in the literature to date, is here provided: With the help of an asymptotic theory, compact expressions for the variance of the estimate of the population total are derived together with variance estimates. These formulas are applicable for moderate values of $N$. The reduction in variance, as compared to sampling with p.p.s. with replacement, is clearly demonstrated.