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A new variance estimator was suggested implicitly by Durbin (1967), for use in a particular multistage sampling procedure. The suggestion can easily be extended to cover the general case in which selection at the first stage is without replacement (the probabilities of inclusion in sample being arbitrary) and in which unbiased estimators are used at all stages of sampling. As compared with the earlier Durbin multistage variance estimator, the new estimator has this advantage; that the conditional expectation (given any particular first-stage sample selection) of the contribution to the estimate of total variance from the second and lower stages is equal to the conditional contribution from these stages to the total variance itself. As a result of this advantage, the new estimator will nearly always be more efficient than the old. The results of a numerical example, however, suggest that the gains in efficiency are unlikely to assume large proportions, at least where the Horvitz-Thompson estimator of total is used. Formulae are also given for use with Des Raj, Murthy, and Rao, Hartley and Cochran estimators.
Journal of the Royal Statistical Society. Series B (Methodological) © 1970 Royal Statistical Society