Bayesian inference for the finite population total from a heteroscedastic probability proportional to size sample

We study Bayesian inference for the population total in probability-proportional-to-size (PPS) sampling. The sizes of non-sampled units are not required for the usual Horvitz-Thompson or Hajek estimates, and this information is rarely included in public use data files. Zheng and Little (2003) showed that including the non-sampled sizes as predictors in a spline model can result in improved point estimates of the finite population total. In Little and Zheng (2007), the spline model is combined with a Bayesian bootstrap (BB) model for the sizes, for point estimation when the sizes are only known for the sampled units. We further develop their methods by (a) including an unknown parameter to model heteroscedastic error variance in the spline model, an important modeling feature in the PPS setting; and (b) developing an improved Bayesian method for including summary information about the aggregate size of non-sampled units. Simulation studies suggest that the resulting Bayesian method, which includes information on the number and total size of the non-sampled units, recovers most of the information in the individual sizes of the non-sampled units, and provides significant gains over the traditional Horvitz-Thompson estimator. The method is applied on a data set from the US Census Bureau.