  Empirical best prediction (EBP) is a well-known method for producing reliable
proportion estimates when the primary data source provides only small or no
sample from finite populations. There are at least two potential challenges
encountered in implementing the existing EBP methodology. First, one must
accurately link the sample to the finite population frame. This may be a
difficult or even impossible task because of absence of identifiers that can be
used to link sample and the frame. Secondly, the finite population frame
typically contains limited auxiliary variables, which may not be adequate for
building a reasonable working predictive model. We propose a data linkage
approach in which we replace the finite population frame by a big sample that
does not have the outcome binary variable of interest, but has a large set of
auxiliary variables. Our proposed method calls for fitting the assumed model
using data from the smaller sample, imputing the outcome variable for all the
units of the big sample, and then finally using these imputed values to obtain
standard weighted proportion using the big sample. We develop a new adjusted
maximum likelihood method to avoid estimates of model variance on the boundary
encountered in the commonly used in maximum likelihood estimation method. We
propose an estimator of mean squared prediction error (MSPE) using a parametric
bootstrap method and address computational issues by developing efficient EM
algorithm. We illustrate the proposed methodology in the context of election
projection for small areas.
