Distribution of Rankings for Groups Exhibiting Heteroscedasticity and Correlation

In one-way analysis of variance, a main interest is in differences among the groups that comprise the population. For a given parameter, such as a mean value, data yield parameter estimates for each group, as well as group rankings based on these statistics. Here ranking probabilities are studied under the assumption that parameter estimates are well approximated by a normal distribution, either in finite samples or asymptotically, with possible intergroup heteroscedasticity and correlation. Particular interest lies in the ranking distribution as a descriptor of experiments on some future dataset. Examples include contract bidding, global economic competition, and rival contests in mating. Ranking distributions are analytically complicated, yet some interesting properties can be derived for them via the symmetry and elliptical geometry of the normal distribution. Some relationships between ranking probabilities and group parameters are described, with attention given to the role of between-group heterogeneity and correlation. For the ranking probabilities, estimators and asymptotically valid standard errors formulas and hypothesis tests are proposed. Simulation is used to describe the sample size needed for accurate asymptotic approximation, and the methods are illustrated with an economic example.