Efficient and accurate inference for mixtures of Mallows models with Spearman distance

The Mallows model(MM) occupies a central role in parametric modelling of ranking data to learn preferences of a population of judges. Despite the wide range of metrics for rankings that can be considered in the model specification, the choice is typically limited to the Kendall, Cayley or Hamming distances, due to the closed-form expression of the related model normalizing constant. This work instead focuses on the Mallows model with Spearman distance (MMS). A novel approximation of the normalizing constant is introduced to allow inference even with a large number of items. This allows us to develop and implement an efficient and accurate EM algorithm for estimating finite mixtures of MMS aimed at i) enlarging the applicability to samples drawn from heterogeneous populations, and ii) dealing with partial rankings affected by diverse forms of censoring. These novelties encompass the critical inferential steps that traditionally limited the use of this distance in practice, and render the MMS comparable (or even preferable) to the MMs with other metrics in terms of computational burden. The inferential ability of the EM scheme and the effectiveness of the approximation are assessed by extensive simulation studies. Finally, we show that the application to three real-world datasets endorses our proposals also in the comparison with competing mixtures of ranking models.