A new robust Bayesian small area estimation via α -stable model for estimating the proportion of athletic students in California.

In the last few years, diabetes mellitus and obesity revealed to be one of the fastest-growing chronic diseases in youth in the United States. The number of new diabetes cases is dramatically increasing, and, for the moment, effective therapy does not exist. Experts believe that one of the causes of this increase is the decline in exercise behavior. The California Education Code requires local educational agencies (LEAs) to administer the FITNESSGRAM, the Physical Fitness Test (PFT), to Californian students of public schools. This test evaluates six fitness areas, and experts defined that a passing result on all six areas of the test represents a fitness level that offers some protection against the diseases associated with physical inactivity. We consider 2015-2016 data provided by the California Department of Education (CDE): for each Californian county ( m=57 ), we aim at estimating the county-level proportion of students with a score equal to six. To account for the heterogeneity of the phenomenon and the presence of outlying counties, we extend the standard area-level model by specifying the random effects as a symmetric α -stable (S α S) distribution that can accommodate different types of outlying observations. The model can accurately estimate the county-level proportion of students with a score equal to six. Results highlight some interesting relationships with social and economic situations in each county. The performance of the proposed model is also investigated through an extensive simulation study.