  Traditional direct estimation methods are not efficient for domains of a
survey population with small sample sizes. To estimate the domain proportions,
we combine the direct estimators and the regression-synthetic estimators based
on domain-level auxiliary information. For the case of small true proportions,
we introduce the design-based linear combination that is a robust alternative
to the empirical best linear unbiased predictor (EBLUP) based on the
Fay--Herriot model. We also consider an adaptive procedure optimizing a
sample-size-dependent composite estimator, which depends on a single parameter
for all domains. We imitate the Lithuanian Labor Force Survey, where we
estimate the proportions of the unemployed and employed in municipalities. We
show where the considered design-based compositions and estimators of their
mean square errors are competitive for EBLUP and its accuracy estimation.
