Hybrid Fay-Herriot Random Forest for Small Area Estimation of Per Capita Expenditure in Bandung Regency

Nabillah Rahmatiah Tangke (Statistics and Data Science Study Program, IPB University)
Asep Saefuddin (Statistics and Data Science Study Program, IPB University)
Anang Kurnia (Statistics and Data Science Study Program, IPB University)

JTAM (Jurnal Teori dan Aplikasi Matematika)
Vol 10, No 4 (2026): October

Direct survey estimates for small areas tend to be too noisy to use on their own when only 15 to 20 households are sampled per area, which is the situation at the sub-district level in Bandung Regency. The Fay-Herriot model in Small Area Estimation (SAE) helps by borrowing strength across areas, but it leans on two assumptions that do not hold well for sub-district per capita expenditure: linearity in the auxiliary variables, and normality of the sampling errors and the area random effects. In practice, PODES indicators relate to expenditure through thresholds and interactions rather than in a straight line, and per capita expenditure at the sub-district level is right-skewed. Both are problems for the classic Fay-Herriot model, and we address them in a single model. Methodologically, it is a quantitative, model-based comparative study that aims to develop and evaluate a hybrid area-level estimator that relaxes the linearity assumption of the Fay-Herriot model and to assess its accuracy for estimating sub-district per capita expenditure. The linear fixed effect is replaced by a Random Forest, yielding a hybrid Fay-Herriot Random Forest (FH-RF), and the response is log-transformed with a Slud and Maiti style bias correction when predictions are mapped back to the original scale. The model is applied to average weekly per capita expenditure on food, beverages, and tobacco across all 31 sub-districts of Bandung Regency, with the response drawn from the 2023 Human Development Index Survey and ten auxiliary variables from PODES. Eight scenarios were compared, varying the model type, the response transformation, and the source of sampling variance. The hybrid FH-RF Log-Normal model with GVF-smoothed variance performed best, with a mean RRMSE of 7.14%, against 9.05% for the strongest classic Fay-Herriot model and 12.75% for the direct estimate, a 44.0% improvement over direct estimation and 21.1% over the best classic specification. 28 of the 31 sub-districts (90.3%) fell below 10% RRMSE, indicating that the hybrid estimator delivers consistently reliable sub-district welfare estimates for poverty targeting.