List Sequential Sampling with Equal or Unequal Probabilities Without Replacement

The problem of selecting a sample of size n from a population of size N with given but unequal selection probabilities πi; i = 1, 2, .., N; and with joint selection probabilities πij such that π i π j — π ij is guaranteed to be non‐negative is one that, while appearing at first sight to be simple, has so far evaded a simple solution. Indeed for n > 2 it has evaded any exact solution. The present paper demonstrates that a simple solution, applicable to a wide variety of the cases that occur in practice, does exist. This solution may require the sampler either (1) to change some selection probabilities, or (2) to allow some variation in sample size. In the first case it is suggested that the changes in many cases will not be serious and may even be beneficial. In the second case the variance of the sample size can be shown to be small. The method proposed is particularly convenient for the sequential selection of units by computer from a list maintained in machine‐readable form.