Robust Bayesian small area estimation using the sub-Gaussian \(\alpha\)-stable distribution for measurement error in covariates

In small area estimation, the sample size is so small that direct estimators have seldom enough adequate precision. Therefore, it is common to use auxiliary data via covariates and produce estimators that combine them with direct data. Nevertheless, it is not uncommon for covariates to be measured with error, leading to inconsistent estimators. Area-level models accounting for measurement error (ME) in covariates have been proposed, and they usually assume that the errors are an i.i.d. Gaussian model. However, there might be situations in which this assumption is violated especially when covariates present severe outlying values that cannot be cached by the Gaussian distribution. To overcome this problem, we propose to model the ME through sub-Gaussian \(\alpha\) $\alpha$ -stable (SG \(\alpha\) $\alpha$ S) distribution, a flexible distribution that accommodates different types of outlying observations and also Gaussian data as a special case when \(\alpha =2\) $\alpha =2$ . The SG \(\alpha\) $\alpha$ S distribution is a generalization of the Gaussian distribution that allows for skewness and heavy tails by adding an extra parameter, \(\alpha \in (0,2]\) $\alpha \in (0,2]$ , to control tail behaviour. The model parameters are estimated in a fully Bayesian framework. The performance of the proposal is illustrated by applying to real data and some simulation studies.