Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications

The concept of \textit{median/consensus} has been widely investigated in order to provide a statistical summary of ranking data, \textit{i.e.} realizations of a random permutation $\Sigma$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$'s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogues of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of \textit{depth function} on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of \textit{consensus ranking} (\textit{medians} corresponding then to the \textit{deepest} permutations). The axiomatic properties that \textit{ranking depths} should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.