  Sample surveys are widely used to obtain information about totals, means,
medians, and other parameters of finite populations. In many applications,
similar information is desired for subpopulations such as individuals in
specific geographic areas and socio-demographic groups. When the surveys are
conducted at national or similarly high levels, a probability sampling can
result in just a few sampling units from many unplanned subpopulations at the
design stage. Cost considerations may also lead to low sample sizes from
individual small areas. Estimating the parameters of these subpopulations with
satisfactory precision and evaluating their accuracy are serious challenges for
statisticians. To overcome the difficulties, statisticians resort to pooling
information across the small areas via suitable model assumptions,
administrative archives, and census data. In this paper, we develop an array of
small area quantile estimators. The novelty is the introduction of a
semiparametric density ratio model for the error distribution in the unit-level
nested error regression model. In contrast, the existing methods are usually
most effective when the response values are jointly normal. We also propose a
resampling procedure for estimating the mean square errors of these estimators.
Simulation results indicate that the new methods have superior performance when
the population distributions are skewed and remain competitive otherwise.
