Hierarchical baxes predictive means and variances with application to sample survey inference

Abstract

Samples are observed from k populations having means ,(i and distributed according to a natural exponential family with quadratic variance function (NEF-QVF). Assuming an exchangeable two-stage prior, conjugate at the first stage, formulas for posterior means, variances and covariances of the μ_i are expressed in a unified way for all NEF-QVF models in terms of integrals over the posterior density of the two hyperparameters. This posterior density is expressed simply in terms of the normalizing function for the conjugate prior.Posterior predictive means and variances for averages of new observations are also obtained. These formulas are applied, with very little effort, to the finite population situation where the samples are independent simple random samples from the k populations, and where an exchangeable NEF-QVF superpopulation model is assumed. Posterior (predictive) means and variances for finite population totals, means and proportions are obtained. These results are illustrated with an application to small area estimation.

Keywords:

• simultaneous estimation
• exchangeability
• posterior means and variances
• exponential families
• bayesian superpopulation models
• small area estimation