Flexible distributions for triple-goal estimates in two-stage hierarchical models.

Performance evaluations often aim to achieve goals such as obtaining estimates of unit-specific means, ranks, and the distribution of unit-specific parameters. The Bayesian approach provides a powerful way to structure models for achieving these goals. While no single estimate can be optimal for achieving all three inferential goals, the communication and credibility of results will be enhanced by reporting a single estimate that performs well for all three. Triple goal estimates [Shen and Louis, 1998. Triple-goal estimates in two-stage hierarchical models. J. Roy. Statist. Soc. Ser. B 60, 455-471] have this performance and are appealing for performance evaluations. Because triple-goal estimates rely more heavily on the entire distribution than do posterior means, they are more sensitive to misspecification of the population distribution and we present various strategies to robustify triple-goal estimates by using nonparametric distributions. We evaluate performance based on the correctness and efficiency of the robustified estimates under several scenarios and compare empirical Bayes and fully Bayesian approaches to model the population distribution. We find that when data are quite informative, conclusions are robust to model misspecification. However, with less information in the data, conclusions can be quite sensitive to the choice of population distribution. Generally, use of a nonparametric distribution pays very little in efficiency when a parametric population distribution is valid, but successfully protects against model misspecification.