A unified analysis of likelihood-based estimators in the Plackett--Luce model

TL;DR

This paper provides a comprehensive asymptotic analysis of likelihood-based estimators in the Plackett--Luce model, including consistency and normality, under various sampling scenarios and graph topologies.

Contribution

It offers the first unified asymptotic theory for different likelihood estimators in the Plackett--Luce model, covering general graph conditions and practical sampling models.

Findings

Establishes uniform consistency of estimators.

Proves asymptotic normality under broad conditions.

Analyzes trade-offs between efficiency and computational complexity.

Abstract

The Plackett--Luce model has been extensively used for rank aggregation in social choice theory. A central statistical question in this model concerns estimating the utility vector that governs the model's likelihood. In this paper, we investigate the asymptotic theory of utility vector estimation by maximizing different types of likelihood, such as full, marginal, and quasi-likelihood. Starting from interpreting the estimating equations of these estimators to gain some initial insights, we analyze their asymptotic behavior as the number of compared objects increases. In particular, we establish both uniform consistency and asymptotic normality of these estimators and discuss the trade-off between statistical efficiency and computational complexity. For generality, our results are proven for deterministic graph sequences under appropriate graph topology conditions. These conditions are shown to be informative when applied to common sampling scenarios, such as nonuniform random hypergraph models and hypergraph stochastic block models. Numerical results are provided to support our findings.