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Multivariate hypergeometric distribution arises frequently in elementarystatistics and probability courses, for simultaneously studying the occurencelaw of specified events, when sampling without replacement from a finitepopulation with fixed number of classification. Covariance matrix of thisdistribution is well known to be identical to its multinomial counterpartmultiplied by 1-(n-1)/(N-1), with N and n being population and sample sizes,respectively. It appears to however, have been less discussed in the literatureabout the meaning of this relationship, especially regarding the specific formof the multiplier. Based on an augmenting argument together with probabilisticsymmetry, we present a more transparent understanding for the covariancestructure of the multivariate hypergeometric distribution. We discussimplications of these combined techniques and provide a unified descriptionabout the relative efficiency for estimating population mean based on simplerandom sampling, probability proportional-to-size sampling and adaptive clustersampling, with versus without replacement. We also provide insight into theclassic random group method for variance estimation.

Better understanding of the multivariate hypergeometric distribution with implications in design-based survey sampling