A Sampling Design for Ordered Populations

We present a sampling algorithm for an ordered population with elements that have a measure of size. The algorithm enables one to select a sample with specified probabilities and with efficiency for the estimated mean between that of one per stratum and that of two per stratum. The algorithm contains a design parameter for the efficiency of the estimated mean relative to the efficiency of the estimated variance of the estimated mean. For a variable highly correlated with the order, it is possible for both the efficiency of the estimated mean and the efficiency of the estimated variance for an intermediate design to be greater than that for the two-per-stratum design. For most studied populations, the variance of the estimated mean declines and the variance of the estimated variance increases as one moves from the two-per-stratum design toward the one-per-stratum design. We illustrate the trade-off between the variance of the estimated mean and the variance of the estimated variance using an autoregressive process. An estimator of the variance of the estimated mean and a replication form for variance estimation are given.