A Beta–Beta Prime Model for Rates and their Precision for Small Area Estimation: An Application to Brazilian Food Insecurity Index

Abstract Statistics bureaus around the world have been faced with an increasing need to provide reliable estimates of economic and social indices, such as proportions or rates, from socioeconomic survey data at small area levels. For example, in 2015, the Member States of the United Nations committed themselves to the 2030 Agenda for Sustainable Development Goals, which requires the national statistical offices and other government agencies of the Member States to provide high-quality, timely, and reliable national indicators at a disaggregated level. However, due to the relatively small sample size of these areas or domains, it is not viable to obtain estimates with an acceptable level of accuracy without using model-based approaches. Here, we propose to model the direct estimator of the rates or proportions in small area domains as being beta distributed. The novelty is that we also model the sampling precision estimator as a beta prime distribution. An evaluation study with real data shows that there is an additional gain in jointly modeling the direct estimator and its sampling precision estimator. An application is also provided to estimate the food insecurity index in small areas of a Brazilian state.