Methods of cluster sampling with and without replacement for clusters of unequal sizes

When the units of a population fall into clusters of different sizes, in such a way that the most economical method of sampling is to record all units, or a subsample of units, within each member of a sample of clusters, the requirements of unbiasedness and precision conflict. Standard methods tend to give either biased estimates with low variances, or unbiased estimates with rather high variances. A method that gives unbiased estimates with relatively low variances is described and compared with existing methods.