A generative model for rank data based on insertion sort algorithm

An original and meaningful probabilistic generative model for full rank data modelling is proposed. Rank data arise from a sorting mechanism which is generally unobservable for statisticians. Assuming that this process relies on paired comparisons, the insertion sort algorithm is known as being the best candidate in order to minimize the number of potential paired misclassifications for a moderate number of objects to be ordered. Combining this optimality argument with a Bernoulli event during a paired comparison step, a model that possesses desirable theoretical properties, among which are unimodality, symmetry and identifiability is obtained. Maximum likelihood estimation can also be performed easily through an EM or a SEM–Gibbs algorithm (depending on the number of objects to be ordered) by involving the latent initial presentation order of the objects. Finally, the practical relevance of the proposal is illustrated through its adequacy with several real data sets and a comparison with a standard rank data model.