Models for Distributions on Permutations

Abstract

A parametric distribution on permutations of k objects is derived from gamma random variables. The probability of a permutation is set equal to the probability that k independent gamma random variables with common shape parameter and different scale parameters are ranked according to that permutation. This distribution is motivated by considering a competition in which k players, scoring points according to independent Poisson processes, are ranked according to the time until r points are scored. The distributions obtained in this way include the popular Luce-Plackett and Thurstone-Mosteller-Daniels ranking models. These gamma-based distributions can serve as alternatives to the null ranking model in which all permutations are equally likely. Here, the gamma models are used to estimate the probability distribution of the order of finish in a horse race when only the probability of finishing first is given for each horse. Gamma models with shape parameters larger than 1 are found to be superior to the most commonly applied model (shape parameter 1). Examples are limited to small values of k because of the complicated calculations required to compute the distribution. Approximations that are easier to calculate are required before more extensive applications can be undertaken.

Key Words:

• Gamma random variable
• Nonnull ranking distribution
• Paired comparisons
• Ranking model