Bayesian Statistical Inference for Sampling a Finite Population

Abstract

Bayesian statistical inference for sampling a finite population is studied by using the Dirichlet-multinomial process as prior. It is shown that if the finite population variables have a Dirichlet-multinomial prior, then the posterior distribution of the inobserved variables given a sample is also Dirichlet-multinomial. If the population size tends to infinity (the sample size is fixed), sampling without replacement from a Dirichlet multinomial process is equivalent to the iid sampling from a Dirichlet process. If both the population size and sample size tend to infinity, then given a sample, the posterior distribution of the population empirical distribution function converges in distribution to a Brownian bridge. The large-sample Bayes confidence band interval are given and shown to be equivalent to the usual ones obtained from simple random sampling.