Sampling with Random Stratum Boundaries

FOR populations arranged in natural order, say in increasing values of a concomitant variable, one common sampling scheme is to divide the population into strata and sample proportionately from each stratum. Variance of the sample mean is minimized (with the possible exception of finite corrections) if the population is divided into n strata and one unit selected from each. A second common procedure, particularly if the sampling is with unequal probabilities, is to sample systematically.t It is well known that if the y characteristic is composed of a linear trend plus random elements the 1-per-stratum design is more efficient than systematic sampling of the population in natural order. The disadvantage of both of these sampling schemes is, of course, that no unbiased estimator of variance is available. In this paper we develop a sampling procedure which for n > 4 and a linear trend has a smaller variance for the sample mean than 1 per stratum and for which an unbiased estimator of the variance is available.