Poisson Counts, Square Root Transformation and Small Area Estimation

The paper intends to serve two objectives. First, it revisits the celebrated Fay-Herriot model, but with homoscedastic known error variance. The motivation comes from an analysis of count data, in the present case, COVID-19 fatality for all counties in Florida. The Poisson model seems appropriate here, as is typical for rare events. An empirical Bayes (EB) approach is taken for estimation. However, unlike the conventional conjugate gamma or the log-normal prior for the Poisson mean, here we make a square root transformation of the original Poisson data, along with square root transformation of the corresponding mean. Proper back transformation is used to infer about the original Poisson means. The square root transformation makes the normal approximation of the transformed data more justifiable with added homoscedasticity. We obtain exact analytical formulas for the bias and mean squared error of the proposed EB estimators. In addition to illustrating our method with the COVID-19 example, we also evaluate performance of our procedure with simulated data as well.