Triple-goal Estimates in Two-stage Hierarchical Models

TLDR

The Bayesian framework excels at structuring complex models and optimizing posterior means under squared error loss, yet these means often fail to accurately estimate parameter histograms or rankings. The study aims to develop a set of estimates that balance ranking accuracy, histogram fidelity, and coordinate‑specific precision, recognizing that no single estimator can simultaneously optimize all three. Three candidate estimators were compared: posterior means, constrained Bayes estimates of Louis and Ghosh, and a novel method that jointly optimizes histogram and rank estimation. Analyses and simulations demonstrate that the novel estimator outperforms the alternatives, achieving superior performance across ranking, histogram, and coordinate‑specific goals.

Abstract

Summary The beauty of the Bayesian approach is its ability to structure complicated models, inferential goals and analyses. To take full advantage of it, methods should be linked to an inferential goal via a loss function. For example, in the two-stage, compound sampling model the posterior means are optimal under squared error loss. However, they can perform poorly in estimating the histogram of the parameters or in ranking them. ‘Triple-goal’ estimates are motivated by the desire to have a set of estimates that produce good ranks, a good parameter histogram and good co-ordinate-specific estimates. No set of estimates can simultaneously optimize these three goals and we seek a set that strikes an effective trade-off. We evaluate and compare three candidate approaches: the posterior means, the constrained Bayes estimates of Louis and Ghosh, and a new approach that optimizes estimation of the histogram and the ranks. Mathematical and simulation-based analyses support the superiority of the new approach and document its excellent performance for the three inferential goals.