In this article, we propose a framework for small area estimation with multiply imputed survey data. Many statistical surveys suffer from (a) high nonresponse rates due to sensitive questions and response burden and (b) too small sample sizes to allow for reliable estimates on (unplanned) disaggregated levels due to budget constraints. One way to deal with missing values is to replace them by several plausible/imputed values based on a model. Small area estimation, such as the model by Fay and Herriot, is applied to estimate regionally disaggregated indicators when direct estimates are imprecise. The framework presented tackles simultaneously multiply imputed values and imprecise direct estimates. In particular, we extend the general class of transformed Fay-Herriot models to account for the additional uncertainty from multiple imputation. We derive three special cases of the Fay-Herriot model with particular transformations and provide point and mean squared error estimators. Depending on the case, the mean squared error is estimated by analytic solutions or resampling methods. Comprehensive simulations in a controlled environment show that the proposed methodology leads to reliable and precise results in terms of bias and mean squared error. The methodology is illustrated by a real data example using European wealth data.
