16 – Bayesian Analysis of Complex Survey Data
Bayesian Inference for the Finite Population Total from a Heteroscedastic Probability Proportional to Size Sample
Roderick Little
University of Michigan/U.S. Census Bureau
Sahar Zangeneh
University of Michigan
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 �les. 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 �nite
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 signi�cant gains over the traditional Horvitz-Thompson estimator. The method is applied
on a data set from the US Census Bureau.
