In this paper a Bayesian least squares approximation is proposed for descriptive inference in a finite population when a categorical auxiliary variable is known. A hierarchical model II analysis of variance is assumed. The solution consists of a projection on the vector of group totals and on the between and within sums of squares. The approximation can be seen as a normal approximation of the joint distribution of the statistic and of the parameter of interest, conditional on two discrete variables: these denote attribution to one group and selection in the sample. The main consequence of the conditioning is that conjectures about non-normality can be taken into account. The solution is compared with others that have been published.
Cocchi, D., & Mouchart, M. (1996). Quasi-linear bayes estimation in stratified finite populations. Royal Statistical Society. Journal. Series B: Methodological, 58(1), 293-300. https://hdl.handle.net/2078.5/144265 (Original work published 1996)