Inference under representable priors for Pearson type II models in finite populations

In this paper we discuss invariant prediction in finite populations. It is assumed that the distribution of the observable quantities is invariant under an orthogonal group of transformations. The quantities of interest are introduced as operational parameters, which depend only on observable quantities. Interest centers on the population total and on the finite population regression coefficient although predictors for the finite population variance are also considered. An operational likelihood function is defined which is a function of the operational parameters. Bayes estimators for the operational parameters are obtained by using the operational likelihood under representable prior distributions yielding conjugate and noninformative distributions. As shown, the Pearson type II distribution plays an important role in deriving the main results.