Empirical Bayesian regression model for estimation of small rates

The efficiency of adding an auxiliary regression variable to the logit model in estimation of small probabilities in large populations is considered. Let us consider two models of distribution of unknown probabilities: the probabilities have gamma distribution (model (A)), or logits of the probabilities have Gaussian distribution (model (B)). In modification of model (B) we will use additional regression variable for Gaussian mean (model (BR)). We have selected real data from Database of Indicators of Statistics Lithuania – Working-age persons recognized as disabled for the first time by administrative territory, year 2010 (number of populations K = 60). Additionally, we have used average annual population data by administrative territory. The auxiliary regression variable was based on data – Number of hospital discharges by administrative territory, year 2010. We obtained initial parameters using simple iterative procedures for models (A), (B) and (BR). At the second stage we performed various tests using Monte-Carlo simulation (using models (A), (B) and (BR)). The main goal was to select an appropriate model and to propose some recommendations for using gamma and logit (with or without auxiliary regression variable) models for Bayesian estimation. The results show that a Monte Carlo simulation method enables us to determine which estimation model is preferable.