Identification of influential values and outliers in the Fay-Herriot EBLUP estimator: Empirical evidence from the crime prevalence in Mexico

Household surveys provide essential data for generating official statistics and informing public policy. However, their sampling designs typically exclude the production of reliable estimates for population subgroups or geographic areas beyond those explicitly targeted. Small area estimation (SAE) techniques are tools that address these limitations, particularly the Empirical Best Linear Unbiased Predictor based on the Fay–Herriot area-level model (EBLUP-FH). Despite being an efficient estimator, is sensitive to outliers, which can breach the statistical assumptions and reduce the accuracy and reliability of results. In this paper, a diagnostic evaluation of the EBLUP-FH was conducted through residual analysis based on robust Mahalanobis and Cook's distances complemented by an analysis of Leverage values. These techniques help to identify areas where the information obtained from the survey and that provided by the auxiliary variables show significant discrepancies compared to the EBLUP-FH predictions. The methods were applied to crime prevalence in Mexican municipalities. The results showed that both methods support the development of consistent and reliable models, with the only notable difference being the number of municipalities identified as outliers. Excluding these areas reduces the influence of extreme observations, thereby preserving the validity of the model's assumptions without compromising the accuracy of the estimates.