Design-based properties of some small-area estimators in forest inventory with two-phase sampling

We consider the small-area estimation problem for forest inventories with two-phase sampling schemes. We propose an improvement to the synthetic estimator, when the true mean of the auxiliary variables over the small area is unknown and must be estimated, and likewise to the residual corrected small-area estimator. We derive the asymptotic design-based variances of these new estimators, the pseudo-synthetic and pseudo-small-area estimators, by also incorporating the design-based variance of the regression coefficients. We then propose a very simple mathematical device that transforms pseudo-small-area estimators into pseudo-synthetic estimators, which is very convenient for deriving asymptotic variances. The results are extended to cluster and two-stage sampling at the plot level. A case study and a simulation illustrate the theory.