  Small area estimation methods are used in surveys, where sample sizes are too
small to get reliable direct estimates of parameters in some population
domains. We consider design-based linear combinations of direct and synthetic
estimators and propose a two-step procedure to approach the optimal
combination. We construct the mean square error estimator suitable for this and
any other linear composition that estimates the optimal one. We apply the
theory to two design-based compositions analogous to the empirical best linear
unbiased predictors (EBLUPs) based on the basic area- and unit-level models.
The simulation study shows that the new methods are efficient compared to
estimation using EBLUP.
