Bayes/EB Ranking, Histogram and Parameter Estimation: Issues and Research Agenda

Abstract

We propose that methods of inference should be linked to an inferential goal via a loss function. For example, if unit-specific parameters are the feature of interest, under squared error loss their posterior means are the optimal estimates. If unit-specific ranks are the target feature (for example to be used in “league tables“, ranking schools, hospitals, physicians or geographic regions), the conditional expected ranks or a discretized version of them are optimal. If the feature of interest is the histogram or empirical distribution function of the unit-specific parameters then the conditional expected edf or a discretized version of it is optimal.

No single set of estimates can simultaneously optimize the three inferential goals. However, in many policy settings communication and credibility will be enhanced by reporting a set of values with good performance for all three goals. This requirement leads to development of “triple-goal” estimates: those producing a histogram that is a good estimate of the parameter histogram, with induced ranks that are good estimates of the parameter ranks and with good performance in estimating unit-specific parameters. Using mathematical and simulation-based analyses, we compare three candidate triple-goal estimates for the two-stage hierarchical model: posterior means, constrained Bayes estimates and a new approach which optimizes estimation of the edf and the ranks.