A nonparametric Bayesian prediction interval for a finite population mean

Given a sample from a finite population, we provide a nonparametric Bayesian prediction interval for a finite population mean when a standard normal assumption may be tenuous. We will do so using a Dirichlet process (DP), a nonparametric Bayesian procedure which is currently receiving much attention. An asymptotic Bayesian prediction interval is well known but it does not incorporate all the features of the DP. We show how to compute the exact prediction interval under the full Bayesian DP model. However, under the DP, when the population size is much larger than the sample size, the computational task becomes expensive. Therefore, for simplicity one might still want to consider useful and accurate approximations to the prediction interval. For this purpose, we provide a Bayesian procedure which approximates the distribution using the exchangeability property (correlation) of the DP together with normality. We compare the exact interval and our approximate interval with three standard intervals, namely the design-based interval under simple random sampling, an empirical Bayes interval and amoment-based interval which uses themean and variance under theDP. However, these latter three intervals do not fully utilize theposterior distributionof the finite populationmeanunder theDP. Using several numerical examples and a simulation study we show that our approximate Bayesian interval is a good competitor to the exact Bayesian interval for different combinations of sample sizes and population sizes. ARTICLE HISTORY Received 31 December 2014 Accepted 2 February 2016