  There has been recent growth in small area estimation due to the need for
more precise estimation of small geographic areas, which has led to groups such
as the U.S. Census Bureau, Google, and the RAND corporation utilizing small
area estimation procedures. We develop novel two-stage benchmarking methodology
using a single weighted squared error loss function that combines the loss at
the unit level and the area level without any specific distributional
assumptions. We consider this loss while benchmarking the weighted means at
each level or both the weighted means and weighted variability at the unit
level. Multivariate extensions are immediate. We analyze the behavior of our
methods using a complex study from the National Health Interview Survey (NHIS)
from 2000, which estimates the proportion of people that do not have health
insurance for many domains of an Asian subpopulation. Finally, the methodology
is explored via simulated data under the proposed model. We ultimately conclude
that three proposed benchmarked Bayes estimators do not dominate each other,
leaving much exploration for future research.
