Credible Distributions of Overall Ranking of Entities

Inference on overall ranking of a set of entities, such as athletes or players, schools and universities, hospitals, cities, restaurants, movies or books, companies, states, countries or subpopulations, based on appropriate characteristics or performances, is an important problem. Estimation of ranks based on point estimates of means does not account for the uncertainty in those estimates. Treating estimated ranks without any regard for uncertainty is problematic. We propose a novel solution using the Bayesian approach. It is easily implementable, competitive with a popular frequentist method, more effective and informative. Using suitable joint credible sets for entity means, we appropriately create {\it credible distributions} (CDs, a phrase we coin), which are probability distributions, for the rank vector of entities. As a byproduct, the supports of the CDs are credible sets for overall ranking. We evaluate our proposed procedure in terms of accuracy and stability using a number of applications and a simulation study. While the frequentist approach cannot utilize covariates, the proposed method handles them routinely to its benefit.