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

TL;DR

This paper introduces a new framework for analyzing ranking data using depth functions on the symmetric group, enabling more comprehensive statistical summaries beyond median-based approaches.

Contribution

It defines novel depth-based concepts for ranking data, extending median and quantile notions to permutations, with theoretical properties and computational methods.

Findings

New depth functions for ranking data are proposed.

Theoretical properties and axioms are established.

Numerical experiments demonstrate practical relevance.

Abstract

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, 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 analogs of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of 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 consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that 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.