Statistical Inference for Incomplete Ranking Data: A Comparison of Two Likelihood-Based Estimators

We consider the problem of statistical inference for ranking data, namely the problem of estimating a probability distribution on the permutation space. Since observed rankings could be incomplete in the sense of not comprising all choice alternatives, we propose to tackle the problem as one of learning from imprecise or coarse data. To this end, we associate an incomplete ranking with its set of consistent completions. We instantiate and compare two likelihood-based approaches that have been proposed in the literature for learning from set-valued data, the marginal and the so-called face-value likelihood. Concretely, we analyze a setting in which the underlying distribution is Plackett-Luce and observations are given in the form of pairwise comparisons.