Estimation of variance in multi-stage sampling

In multi-stage sampling when selection is without replacement at the first stage, estimation of the variance of the estimate of the population total is often done assuming sampling with replacement. This estimate is biased and the degree of bias is not negligible. In this paper, a procedure which gives unbiased estimates of the variance making use of only estimated primary sampling unit totals is suggested for the case when sampling at the second and subsequent stages is simple random without replacement. This procedure is based on sub-samples drawn from the selected second and subsequent stage units.