Three-fold Fay–Herriot model for small area estimation and its diagnostics

This paper introduces a three-fold Fay–Herriot model with random effects at three hierarchical levels. Small area best linear unbiased predictors of linear indicators are derived from the new model and the corresponding mean squared errors are approximated and estimated analytically and by parametric bootstrap. The problem of influence analysis and model diagnostics is addressed by introducing measures adapted to small area estimation. Simulation experiments empirically investigate the behavior of the predictors and mean squared error estimators. The new statistical methodology is applied to Spanish living conditions survey of 2004–2008. The target is the estimation of proportions of women and men under the poverty line by province and year.