To replace or not to replace in finite population sampling

We revisit the classical result in finite population sampling which states that in equally-likely "simple" random sampling the sample mean is more reliable when we do not replace after each draw. In this paper we investigate if and when the same is true for samples where it may no longer be true that each member of the population has an equal chance of being selected. For a certain class of sampling schemes, we are able to obtain convenient expressions for the variance of the sample mean and surprisingly, we find that for some selection distributions a more reliable estimate of the population mean will happen by replacing after each draw. We show for selection distributions lying in a certain polytope the classical result prevails.