Mitigating measurement error in misspecified small area models

In the framework of Small area estimation, we consider an area‐level model where a subset of covariates is measured with error. The extent of the error is assumed to be constant throughout the areas, and it is expressed by a scalar parameter , which multiplies the deterministic covariance matrix of the estimator of the true covariates. In the absence of relevant information, imposing a fully deterministic covariance structure of the measurement error may lead to model misspecification. Small area models naturally include random effects, and a misspecified modelling of potential measurement error might inflate both bias and standard error of the area estimates, providing unreliable indicators. In order to induce an increasing shrinkage toward the potential absence of measurement error as long as we add information through the inclusion of new covariates, we adopt an ad hoc defined penalized complexity prior on . We claim that our proposal provides more robust estimates with respect to the case of a deterministic covariance structure of the measurement error. We also provide conditions for the posterior propriety.