We consider the adjustment, based upon a sample of size $n$, of collections of vectors drawn from either an infinite or finite population. The vectors may be judged to be either Normally distributed or, more generally, second-order exchangeable. We develop the work of Goldstein and Wooff (1998) to show how the familiar univariate finite population corrections (fpc) naturally generalise to individual quantities in the multivariate population. The types of information we gain by sampling are identified with the orthogonal canonical variable directions derived from a generalised eigenvalue problem. These canonical directions share the same co-ordinate representation for all sample sizes and, for equally defined individuals, all population sizes enabling simple comparisons between both the effects of different sample sizes and of different population sizes. We conclude by considering how the fpc is modified for multivariate cluster sampling with exchangeable clusters. In univariate two-stage cluster sampling we may decompose the variance of the population mean into the sum of the variance of cluster means and the variance of the cluster members within clusters. The first term has a fpc relating to the sampling fraction of clusters, the second term has a fpc relating to the sampling fraction of cluster size. We illustrate how this generalises in the multivariate case. We decompose the variance into two terms: the first relating to multivariate finite population sampling of clusters and the second to multivariate finite population sampling within clusters. We solve two generalised eigenvalue problems to show how to generalise the univariate to the multivariate: each of the two fpcs attaches to one, and only one, of the two eigenbases.
Shaw, S. C., & Goldstein, M. (2012). Finite population corrections for multivariate Bayes sampling. Journal of Statistical Planning and Inference, 142(10), 2844-2861. https://doi.org/10.1016/j.jspi.2012.04.001