  In small area estimation different data sources are integrated in order to
produce reliable estimates of target parameters (e.g., a mean or a proportion)
for a collection of small subsets (areas) of a finite population. Regression
models such as the linear mixed effects model or M-quantile regression are
often used to improve the precision of survey sample estimates by leveraging
auxiliary information for which means or totals are known at the area level. In
many applications, the unit-level linkage of records from different sources is
probabilistic and potentially error-prone. In this paper, we present
adjustments of the small area predictors that are based on either the linear
mixed effects model or M-quantile regression to account for the presence of
linkage error. These adjustments are developed from a two-component mixture
model that hinges on the assumption of independence of the target and auxiliary
variable given incorrect linkage. Estimation and inference is based on
composite likelihoods and machinery revolving around the
Expectation-Maximization Algorithm. For each of the two regression methods, we
propose modified small area predictors and approximations for their mean
squared errors. The empirical performance of the proposed approaches is studied
in both design-based and model-based simulations that include comparisons to a
variety of baselines.
