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

Ruijian Han, Yiming Xu

TL;DR

This work develops a unified asymptotic theory for likelihood-based estimators in the Plackett-Luce model under general hypergraph comparison graphs. It analyzes full, marginal MLE, and quasi-likelihood estimators, proving uniform consistency for deterministic graphs and, under balancing conditions, asymptotic normality for both random (NURHM, HSBM) and deterministic designs. The theory introduces rapid expansion and edge-sharing topology notions to connect graph structure with estimator variance, and it demonstrates trade-offs between statistical efficiency and computational cost. Numerical experiments, including a horse-racing dataset, validate the theory and show practical uncertainty quantification across nonuniform, heterogeneous graphs.

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.

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

TL;DR

This work develops a unified asymptotic theory for likelihood-based estimators in the Plackett-Luce model under general hypergraph comparison graphs. It analyzes full, marginal MLE, and quasi-likelihood estimators, proving uniform consistency for deterministic graphs and, under balancing conditions, asymptotic normality for both random (NURHM, HSBM) and deterministic designs. The theory introduces rapid expansion and edge-sharing topology notions to connect graph structure with estimator variance, and it demonstrates trade-offs between statistical efficiency and computational cost. Numerical experiments, including a horse-racing dataset, validate the theory and show practical uncertainty quantification across nonuniform, heterogeneous graphs.

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.
Paper Structure (56 sections, 20 theorems, 272 equations, 2 figures, 4 tables)

This paper contains 56 sections, 20 theorems, 272 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Organization
  5. Notation
  6. Problem Set-up
  7. The PL Model
  8. Comparison Data
  9. Comparison Graphs
  10. Nonuniform Random Hypergraph Models
  11. Hypergraph Stochastic Block Models
  12. Utility Estimation
  13. The Likelihood Approach
  14. Full Observations
  15. Choice-one Observations
  16. ...and 41 more sections

Key Result

Proposition 3.1

The MLE $\widehat{\bm{u}}$ obtained from maximizing mmle subject to $\langle 1\rangle^\top\bm{u} = 0$, if exists, satisfies where and $\mathscr F_{i, j-1}$ are defined in filt. Meanwhile, the true parameter $\bm{u}^*$ satisfies $\mathbb E[S_i(\bm u^*; k)] = 0$ for every $i\in [N]$.

Figures (2)

  • Figure 1: Average $\ell_\infty$ error with corresponding quartiles for the marginal MLE (the choice-one MLE and the choice-2 MLE), the full MLE, and the QMLE and the frequencies of each estimator yielding the smallest $\ell_\infty$ error among others in $300$ experiments as the number of objects $n$ increases from $200$ to $1000$. (a)-(b): NURHM; (c)-(d): HSBM.
  • Figure 2: Coverage probabilities when adding additional data to increase the heterogeneity of an HSBM. (a): Full MLE; (b): QMLE.

Theorems & Definitions (31)

  • Proposition 3.1
  • Remark 1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 4.1: Uniform consistency under NURHM
  • Theorem 4.2: Uniform consistency under HSBM
  • Theorem 4.3: Asymptotic normality of marginal MLE
  • Remark 2
  • Remark 3
  • Theorem 4.4: Asymptotic normality of QMLE
  • ...and 21 more