Four parameter beta GLMM Bayesian inference approach: Improving paddy productivity predictions for area yield index crop insurance

Paddy is a staple crop and a vital component of Indonesia's agriculture, significantly contributing to food security and rural livelihood. Nevertheless, paddy cultivation is highly vulnerable to risks such as pests, diseases, extreme weather, and natural disasters, which can lead to significant productivity losses. Thus, schemes like Area Yield Index (AYI) insurance have a critical role in mitigating these risks because. It provides financial protection to farmers by compensating them for losses due to area-wide productivity shortfalls. Hence, accurate predictions of paddy productivity are essential for setting fair and precise AYI premiums. Therefore, this study proposes an innovative framework by developing a four-parameter beta distribution Generalized Linear Mixed Model (GLMM) based on a Bayesian approach for predicting paddy productivity. The approach is motivated by the model’s ability to apply a four-parameter beta distribution that effectively models the bounded nature of paddy productivity and ensures that predictions remain within realistic range value. The inclusion of random effects also accounts for variability of paddy productivity across areas, which is commonly found in Indonesia and other countries. Meanwhile, the Bayesian framework further enhances robustness by integrating prior knowledge and providing probabilistic predictions. Based on the proposed approach, we then design an enhanced AYI policy based on district and sub district conditions. The framework is first developed through simulation studies designed to replicate real paddy productivity conditions. Comparative testing of the Stan and BRMS packages in R reveals that the proposed four-parameter beta GLMM implemented in Stan is more flexible and accurate. The methodology is then applied to an empirical case study predicting paddy productivity in Central Kalimantan (2020), using farmer survey data and lagged values of Sentinel-2A satellite indices (bands 4, 8, and NDVI) as covariates. Results show that agronomic practices such as pest management and current and historical satellite data enhance prediction accuracy, demonstrating the model's potential to predict productivity with high precision, proving that the proposed method is well-suited for calculating premiums and risks under AYI crop insurance policies. The estimated pure AYI premium ranges from IDR 300.000 to 410.000. Unlike conventional premium calculations based on average historical yields, the proposed GLMM approach provides a nuanced, data-driven alternative that accounts for various productivity factors, ensuring greater adaptability, accuracy, and responsiveness to changes in agricultural conditions, including those driven by climate change.