Outlier robust estimation of a finite population total

We discuss a Bayesian approach to estimating a finite population total when the mean of the observed sample values is assumed to be proportional to a known covariate. Observations have one of two possible variances, a standard form or an inflated form. Observations with the larger variance are termed outliers. We learn about the proportion of outliers and parameters of the model from the sample. Inference about the mean and variance of the population total is made using the predictive distribution. The frequentist properties of the procedure are analyzed for a particular finite population by examining the bias of the estimator and coverage of the confidence interval. The performance of the estimator is shown to be superior to the Horvitz-Thompson estimator and a simple Bayes estimator which allows only the standard variance.