Author: Shonosuke Sugasawa (Risk Analysis Research Center, The Institute of Statistical Mathematics)
Empirical Bayes estimators are widely used to provide indirect and model-based estimates of means in small areas. The most common model is a two-stage normal hierarchical model called Fay-Herriot model. However, due to the normality assumption, it might be highly influenced by the presence of outliers. In this talk, we propose a simple modification of the conventional method by using density power divergence (Basu et al., 1998). The resulting robust marginal likelihood function based on density power divergence is characterized by a scalar parameter controlling the robustness and the proposed likelihood includes the usual marginal likelihood function as a special case. The Bayes estimator based on the likelihood is tail-robust in the sense that the Bayes estimator performs similarly to the direct estimator in outlying areas whereas the usual Bayes estimator does not have such a property. We also show that the estimator of model parameters based on the proposed likelihood is asymptotically normal, and we derive an second order unbiased estimator of the area-specific mean squared error of the robust empirical Bayes estimator. Through simulation studies, we compare the proposed method with some existing methods. Finally, the proposed method is successfully applied to the milk data available in “sae” package in R.
References:Basu, A., Harris, I. R., Hjort, N. L. and Jones, M. C. (1998). Robust and efficient estimation by minimizing a density power divergence. Biometrika, 85, 549-559.